Search: id:A007318 Results 1-1 of 1 results found. %I A007318 M0082 %S A007318 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,20,15,6,1,1,7,21,35, %T A007318 35,21,7,1,1,8,28,56,70,56,28,8,1,1,9,36,84,126,126,84,36,9,1,1,10,45, %U A007318 120,210,252,210,120,45,10,1,1,11,55,165,330,462,462,330,165,55,11,1 %N A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n. %C A007318 C(n,k) = number of k-element subsets of an n-element set. %C A007318 Row n gives coefficients in expansion of (1+x)^n. %C A007318 C(n+k-1,n-1) is the number of ways of placing k indistinguishable balls into n boxes (the "bars and stars" argument - see Feller). %C A007318 C(n-1,m-1) is the number of compositions of n with m summands. %C A007318 If thought of as an infinite lower triangular matrix, inverse begins: %C A007318 +1 %C A007318 -1 +1 %C A007318 +1 -2 +1 %C A007318 -1 +3 -3 +1 %C A007318 +1 -4 +6 -4 +1 %C A007318 The string of 2^n palindromic binomial coefficients starting after the A006516(n)-th entry are all odd. - Lekraj Beedassy (blekraj(AT)yahoo.com), May 20 2003 %C A007318 C(n+k-1,n-1) is the number of standard tableaux of shape (n,1^k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004 %C A007318 Can be viewed as an array, read by antidiagonals, where the entries in the first row and column are all 1's and A(i,j) = A(i-1,j) + A(i, j-1) for all other entries. The determinants of all its n X n subarrays starting at (0,0) are all 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 17 2004 %C A007318 Also the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals j+1 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006 %C A007318 C(n-3,k-1) counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 132 and k descents. C(n-3,k-1) also counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 213 and k descents. - David Hoek (david.hok(AT)telia.com), Feb 28 2007 %C A007318 Inverse of A130595 (as an infinite lower triangular matrix). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 21 2007 %C A007318 Consider integer lists LL of lists L of the form LL=[m#L]=[m#[k#2]] (where '#' means 'times') like LL(m=3,k=3) = [[2,2,2],[2,2,2],[2,2,2]]. The number of the integer list partitions of LL(m,k) is equal to C(m+k,k) if multiple partitions like [[1,1],[2],[2]] and [[2],[2], [1,1]] and [[2],[1,1],[2]] are count only once. For the example we find 4*5*6/3! = 20 = C(6,3). - Thomas Wieder (thomas.wieder(AT)t-online.de), Oct 03 2007 %C A007318 The infinitesimal generator for the Pascal triangle and its inverse is A132440. - Tom Copeland (tcjpn(AT)msn.com), Nov 15 2007 %C A007318 Row n>=2 gives the number of k-digit (k>0) base n numbers with strictly decreasing digits; e.g. row 10 for A009995. Similarly, row n-1>=2 gives the number of k-digit (k>1) base n numbers with strictly increasing digits; see A009993 and compare A118629. - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Nov 25 2007 %C A007318 Comments from Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008: (Start) C(n+k-1, k) is the number of ways a sequence of length k can be partitioned into n subsequences (see the Naish link). %C A007318 C(n+k-1, k) is also the number of n- (or fewer) digit numbers written in radix at least k whose digits sum to k. For example, in decimal, there are C(3+3-1,3)=10 3-digit numbers whose digits sum to 3 (see A052217) and also C(4+2-1,2)=10 4-digit numbers whose digits sum to 2 (see A052216). This relationship can be used to generate the numbers of sequences A052216 to A052224 (and further sequences using radix greater than 10). (End) %C A007318 Denote by sigma_k(x_1,x_2,...,x_n) the elementary symmetric polynomials. Then: C(2n+1,2k+1)=sigma_{n-k}(x_1,x_2,...,x_n), where x_i=tan^2(i*Pi/ (2n+1)),(i=1,2,...,n). C(2n,2k+1)=2n*sigma_{n-1-k}(x_1,x_2,...,x_{n-1}), where x_i=tan^2(i*Pi/(2n)), (i=1,2,...,n-1). C(2n,2k)=sigma_{n-k}(x_1, x_2,...,x_n), where x_i=tan^2((2i-1)Pi/(4n)), (i=1,2,...,n). C(2n+1, 2k)=(2n+1)sigma_{n-k}(x_1,x_2,...,x_n), where x_i=tan^2((2i-1)Pi/ (4n+2)), (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), May 07 2008 %C A007318 Given matrices R and S with R(n,k) = C(n,k)*r(n-k) and S(n,k) = C(n,k)*s(n-k), then R*S = T where T(n,k) = C(n,k)*[r(.)+s(.)]^(n-k), umbrally. And, the e.g.f.s for the row polynomials of R, S and T are, respectively, exp(x*t)*exp[r(.)*x], exp(x*t)*exp[s(.)*x] and exp(x*t)*exp[r(.)*x]*exp[s(.)*x] = exp{[t+r(.)+s(.)]*x}. The row polynomials are essentially Appell polynomials. See A132382 for an example. [From Tom Copeland (tcjpn(AT)msn.com), Aug 21 2008] %C A007318 Contribution from Clark Kimberling (ck6(AT)evansville.edu), Sep 15 2008: (Start) %C A007318 As the rectangle R(m,n)=C(m+n-2,m-1), the weight array W (defined %C A007318 generally at A114112) of R is essentially R itself, in the sense that %C A007318 if row 1 and column 1 of W=A144225 are deleted, the remaining array is R. (End) %C A007318 If A007318 = M as infinite lower triangular matrix, M^n gives A130595, A023531, A007318, A038207, A027465, A038231, A038243, A038255, A027466, A038279, A038291, A038303, A038315, A038327, A133371, A147716, A027467 for n=-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2008] %C A007318 The coefficients of the polynomials with e.g.f. exp(x*t)*(cosh(t)+sinh(t)). [From Peter Luschny (peter(AT)luschny.de), Jul 09 2009] %D A007318 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828. %D A007318 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. %D A007318 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 63ff. %D A007318 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 4. %D A007318 Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7. %D A007318 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 306. %D A007318 P. Curtz, Integration numerique des systemes differentiels..,C.C.S.A., Arcueil,1969. [From Paul Curtz (bpcrtz(AT)free.fr), Mar 06 2009] %D A007318 W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 36, 1968. %D A007318 D. Fowler, The binomial coefficient function, Amer. Math. Monthly, 103 (1996), 1-17. %D A007318 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 155. %D A007318 D. Hoek, Parvisa moenster i permutationer [Swedish], 2007. %D A007318 D. E. Knuth, The Art of Computer Programming, Vol. 1, 2nd ed., p. 52. %D A007318 S. K. Lando, Lecture on Generating Functions, Amer. Math. Soc., Providence, R.I., 2003, pp. 60-61. %D A007318 D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp. %D A007318 Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121. %D A007318 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71. %D A007318 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 6. %D A007318 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2. %D A007318 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996, p. 143. %D A007318 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239. %D A007318 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A007318 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 115-8, Penguin Books 1987. %H A007318 N. J. A. Sloane, First 141 rows of Pascal's triangle, formatted as a simple linear sequence n, a(n) %H A007318 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A007318 V. Asundi, Generate a Yanghui Triangle [Broken link] %H A007318 C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps %H A007318 D. Butler, Pascal's Triangle %H A007318 L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n %H A007318 L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n E709 %H A007318 S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654) %H A007318 Nick Hobson, Python program %H A007318 Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom %H A007318 S. Kak, The Golden Mean and the Physics of Aesthetics %H A007318 W. Knight, Short Table of Binomial Coefficients %H A007318 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. %H A007318 Mathforum, Pascal's Triangle %H A007318 Mathforum, Links for Pascal's triangle %H A007318 C. McDermottroe, n-th row generator of Pascal's triangle %H A007318 D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees %H A007318 Lee Naish Pascal's Triangle and debugging software %H A007318 A. Necer, Series formelles et produit de Hadamard %H A007318 G. Sivek et al., ThinkQuest, Pascal's Triangle Row Generator %H A007318 N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98). %H A007318 H. Verrill, Pascal's Triangle and related triangles %H A007318 G. Villemin's Almanach of Numbers, Triangle de Pascal %H A007318 Eric Weisstein's World of Mathematics, More information. %H A007318 Thomas Wieder, Home Page. %H A007318 Thomas Wieder, (Old) Home Page. %H A007318 Wikipedia, Pascal's triangle %H A007318 H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, pp. 12ff. %H A007318 K. Williams, Mathforum, Interactive Pascal's Triangle %H A007318 K. Williams, MathForum, Pascal's Triangle to Row 19 %H A007318 D. Zeilberger, [math/9809136] The Combinatorial Astrology of Rabbi Abraham Ibn Ezra %H A007318 Index entries for triangles and arrays related to Pascal's triangle %F A007318 a(n, m)=binomial(n, m); a(n+1, m) = a(n, m)+a(n, m-1), a(n, -1) := 0, a(n, m) := 0, n=0); also g.f.: 1/(1-x-y)=Sum(C(n+k, k)x^k*y^n, n, k>=0). G.f. for row n: (1+x)^n = sum(k=0..n, C(n, k)x^k). G.f. for column n: x^n/(1-x)^n. %F A007318 E.g.f.: A(x, y)=exp(x+xy). E.g.f. for column n: x^n*exp(x)/n!. %F A007318 In general the m-th power of A007318 is given by: T(0, 0) = 1, T(n, k) = T(n-1, k-1) + m*T(n-1, k), where n is the row-index and k is the column; also T(n, k) = m^(n-k) C(n, k). %F A007318 Triangle T(n, k) read by rows; given by A000007 DELTA A000007, where DELTA is Deleham's operator defined in A084938. %F A007318 With P(n+1) = the number of integer partitions of (n+1), p(i) = the number of parts of the i-th partition of (n+1), d(i) = the number of different parts of the i-th partition of (n+1), m(i, j) = multiplicity of the j-th part of the i-th partition of (n+1), sum_[p(i)=k]_{i=1}^{P(n+1)} = sum running from i=1 to i=P(n+1) but taking only partitions with p(i)=(k+1) parts into account, prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has B(n, k) = sum_[p(i)=(k+1)]_{i=1}^{P(n+1)} 1/prod_{j=1}^{d(i)} m(i, j)! E.g. B(5, 3) = 10 because n=6 has the following partitions with m=3 parts: (114), (123), (222). For their multiplicities one has: (114): 3!/(2!*1!) = 3, (123): 3!/(1!*1!*1!) = 6, (222): 3!/3! = 1. The sum is 3+6+1=10=B(5, 3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 03 2005 %F A007318 C(n, k) = Sum_{j, 0<=j<=k} = (-1)^j*C(n+1+j, k-j)*A000108(j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 10 2005 %F A007318 G.f.: 1 + x(1 + x) + x^3(1 + x)^2 + x^6(1 + x)^3 + ... . - Michael Somos Sep 16 2006 %F A007318 Sum_{k, 0<=k<=[n/2]} x^(n-k)*T(n-k,k)= A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . Sum_{k, 0<=k<=[n/2]} (-1)^k*x^(n-k)*T(n-k,k)= A000007(n), A010892(n), A009545(n+1), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n), A084329(n+1) for x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 16 2006 %F A007318 C(n,k) <= A062758(n) for n > 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 04 2008 %F A007318 C(t+p-1, t) = Sum(i=0..t, C(i+p-2, i)) = Sum(i=1..p, C(i+t-2, t-1)) A binomial number is the sum of its left parent and all its right ancestors, which equals the sum of its right parent and all its left ancestors. - Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008 %e A007318 Triangle begins: %e A007318 1 %e A007318 1, 1 %e A007318 1, 2, 1 %e A007318 1, 3, 3, 1 %e A007318 1, 4, 6, 4, 1 %e A007318 1, 5, 10, 10, 5, 1 %e A007318 1, 6, 15, 20, 15, 6, 1 %e A007318 1, 7, 21, 35, 35, 21, 7, 1 %e A007318 1, 8, 28, 56, 70, 56, 28, 8, 1 %e A007318 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 %e A007318 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1 %e A007318 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1 %p A007318 A007318 := (n,k)->binomial(n,k); %p A007318 with(combstruct):for n from 0 to 11 do seq(count(Combination(n),size=m), m = 0 .. n) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007 %t A007318 Flatten[Table[Binomial[n, k], {n, 0, 11}, {k, 0, n}]] (from Robert G. Wilson v Jan 19 2004) %o A007318 (AXIOM) -- (start) %o A007318 )set expose add constructor OutputForm %o A007318 pascal(0,n) == 1 %o A007318 pascal(n,n) == 1 %o A007318 pascal(i,j | 0 < i and i < j) == pascal(i-1,j-1) + pascal(i,j-1) %o A007318 pascalRow(n) == [pascal(i,n) for i in 0..n] %o A007318 displayRow(n) == output center blankSeparate pascalRow(n) %o A007318 for i in 0..20 repeat displayRow i -- (end) %o A007318 (PARI) C(n,k)=if(k<0|k>n,0,n!/k!/(n-k)!) %o A007318 (PARI) C(n,k)=if(n<0,0,polcoeff((1+x)^n,k)) %o A007318 (PARI) C(n,k)=if(k<0|k>n,0, if(k==0&n==0,1,C(n-1,k)+C(n-1,k-1))) %o A007318 (Python) See Hobson link. %Y A007318 Equals differences between consecutive terms of A102363 - David G. Williams (davidwilliams(AT)Paxway.com), Jan 23 2006 %Y A007318 Cf. A047999, A026729, A052553. Row sums give A000079 (powers of 2). %Y A007318 Cf. A083093 (triangle read mod 3). %Y A007318 Partial sums of rows give triangle A008949. %Y A007318 Infinite matrix squared: A038207, cubed: A027465 %Y A007318 Cf. A101164. If rows are sorted we get A061554 or A107430. %Y A007318 Another version: A108044. %Y A007318 Cf. A008277. %Y A007318 Cf. A132311, A132312. %Y A007318 Cf. A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223. %Y A007318 Cf. A144225. [From Clark Kimberling (ck6(AT)evansville.edu), Sep 15 2008] %Y A007318 Sequence in context: A154926 A117440 A118433 this_sequence A108086 A130595 A108363 %Y A007318 Adjacent sequences: A007315 A007316 A007317 this_sequence A007319 A007320 A007321 %K A007318 nonn,tabl,nice,easy,core %O A007318 0,5 %A A007318 N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu) Search completed in 0.007 seconds