Search: id:A001787 Results 1-1 of 1 results found. %I A001787 M3444 N1398 %S A001787 0,1,4,12,32,80,192,448,1024,2304,5120,11264,24576,53248,114688,245760, %T A001787 524288,1114112,2359296,4980736,10485760,22020096,46137344,96468992, %U A001787 201326592,419430400,872415232,1811939328,3758096384,7784628224 %N A001787 n*2^(n-1). %C A001787 Number of edges in n-dimensional hypercube. %C A001787 Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2001 %C A001787 Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2 - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001 %C A001787 Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 26 2002 %C A001787 (-1) times determinant of matrix A_{i,j} = -|i-j|, 0<=i,j<=n. %C A001787 a(n)= number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n)-A000337(n-1) for n = 2,3,... - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2003 %C A001787 The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark (eclark(AT)math.usf.edu), May 27 2003 %C A001787 Binomial transform of [0,1,2,3,4,5,...]. Without the initial 0, binomial transform of odd numbers. %C A001787 With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n(n-1)+0^n)/ 4. - Paul Barry (pbarry(AT)wit.ie), May 20 2003 %C A001787 PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial transform is A027471(n+1). - Michael Somos, Jul 10 2003 %C A001787 Number of zeros in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003 %C A001787 Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n(or first n+1 nonnegative integers A001477);Illustrating the case n=5: %C A001787 0...1...2...3...4...5 %C A001787 ..1...3...5...7...9 %C A001787 ....4...8...12..16 %C A001787 ......12..20..28 %C A001787 ........32..48 %C A001787 ..........80 and final element is a(5)=80. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004 %C A001787 This sequence and A001871 arise in counting ordered trees of height at most k where only the right-most branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871. %C A001787 Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004 %C A001787 Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1, j2), a(i2,j1)) where i1(1/sqrt(1-4x^2))g(xc(x^2)), c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 13 2005 %C A001787 Sequences A018215 and A058962 interleaved. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006 %C A001787 The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. - Ben Thurston (benthurston27(AT)yahoo.com), Nov 13 2006 %C A001787 Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. - Ross La Haye (rlahaye(AT)new.rr.com), Dec 30 2006 %C A001787 Convolution of the natural numbers [A000027] and A045623 beginning [0, 1,2,5...]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 03 2007 %C A001787 If M is the matrix (given by rows) [2,-1;0,2] then the sequence gives the (1,2) entry in M^n. - Antonio M. Oller (oller(AT)unizar.es), May 21 2007 %C A001787 If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), Jul 21 2007 %C A001787 Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 27 2007 %C A001787 A member of the family of sequences defined by a(n) = n*[c(1)*...c(r)]^(n-1); c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Feb 23 2008 %C A001787 Sum(n>0,1/a(n))=2log(2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 10 2009] %C A001787 Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3, 6, 12,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009] %C A001787 Starting with offset 1 = A059570: (1, 2, 6, 14, 34,...) convolved with (1, 2, 2, 2,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009] %D A001787 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A001787 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001787 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796. %D A001787 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 131. %D A001787 M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris, eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168. See p. 152. %D A001787 F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. %D A001787 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=-4. %D A001787 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203. %D A001787 T. Y. Lam, On the diagonalization of quadratic forms, Math. Mag., 72 (1999), 231-235 (see page 234). %D A001787 W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38) and (45), lhs, m=4. %D A001787 Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009] %H A001787 Franklin T. Adams-Watters, Table of n, a(n) for n = 0..500 %H A001787 Index entries for sequences related to linear recurrences with constant coefficients %H A001787 Milan Janjic, Two Enumerative Functions %H A001787 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 A001787 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001787 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001787 D. W. Bass and I. H. Sudborough, Hamilton decompositions and (n/2)-factorizations of hypercubes, J Graph Algor. Appl. 7(2003) 79-98. %H A001787 D. Callan, A recursive bijective approach to counting permutations... %H A001787 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A001787 F. Ellermann, Illustration of binomial transforms %H A001787 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 408 %H A001787 S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp. %H A001787 S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns , University of Kentucky Research Reports (2004). %H A001787 M. L. Perez et al., eds., Smarandache Notions Journal %H A001787 A. Robertson, Permutations containing and avoiding 123 and 132 patterns %H A001787 A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6, 1999, #R38. %H A001787 Eric Weisstein's World of Mathematics, Hypercube %H A001787 Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle %H A001787 Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008). %H A001787 Index entries for sequences related to Chebyshev polynomials. %F A001787 a(n) = sum(k=1, n, k*binomial(n, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 06 2002 %F A001787 E.g.f. xexp(2x) - Paul Barry (pbarry(AT)wit.ie), Apr 10 2003 %F A001787 G.f.: x/(1-2x)^2. a(n)=2a(n-1)+2^(n-1). a(2n)= n4^n, a(2n+1)= (2n+1)4^n. %F A001787 Starting 1, 1, 4, 12, .. this is 0^n+n2^(n-1), the binomial transform of the 'pair-reversed' natural numbers A004442 - Paul Barry (pbarry(AT)wit.ie), Jul 24 2003 %F A001787 Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry (perry(AT)globalnet.co.uk), Aug 07 2003 %F A001787 The signed version of this sequence, n(-2)^(n-1), is the inverse binomial transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul Barry (pbarry(AT)wit.ie), Aug 20 2003 %F A001787 a(n-1)=sum{k=0..n, 2^(n-k-1)C(n-k, k)C(1, (k+1)/2)(1-(-1)^k)/2}-0^n/4. - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004 %F A001787 a(n)=sum{k=0..floor(n/2), binomial(n, k)(n-2k)^2}; - Paul Barry (pbarry(AT)wit.ie), May 13 2005 %F A001787 a(n+2) = A049611(n+2) - A001788(n). Floretion Algebra Multiplication Program, FAMP Code: 1vessum(pos)seq[A], 1vessum(neg)seq[A] and 1vessumseq[A] (= (a(n)) from 2nd term) with A = + .5'i + .5i' + .5'ij' + .5'ki' + 2e. Sumtype is set to: default (ver. f) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 02 2005 %F A001787 a(n)=n!sum{k=0..n, 1/((k - 1)!(n - k)!)} - Paul Barry (pbarry(AT)wit.ie), Mar 26 2003 %F A001787 a(n) = sum(binomial(n+1,j)*(n+1-j),j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 22 2006 %F A001787 a(n+1)=Sum_{k, 0<=k<=n}4^k*A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2006 %F A001787 Row sums of A130300 starting (1, 4, 12, 32,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 20 2007 %F A001787 Equals row sums of triangle A134083. Equals A002064(n) + (2^n - 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2007 %F A001787 a(n)=4*a(n-1)-4*a(n-2), a(0)=0, a(1)=1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008] %F A001787 a(n) is the number of ways to split {1,2,...n-1} into two (possibly empty) complimentary intervals {1,2,...i} and {i+1,i+2,...n-1} and then select a subset from each interval. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Jan 31 2009] %e A001787 a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations of 1234 containing exactly one increasing subsequence of length 3. %p A001787 spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 09 2006 %p A001787 a:=n->sum (2^(n-1),j=1..n): seq(a(n),n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007 %p A001787 A001787:=1/(2*z-1)^2; [S. Plouffe in his 1992 dissertation, dropping the initial zero.] %p A001787 with (combinat): c := n -> stirling2(n,2): b := n -> if n<2 then 1; else c(n)-c(n-1); fi: a := n -> add(b(i)*c(n-i), i=1..n-1): seq(a(n),n=2..31); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 10 2007 %p A001787 with(finance):seq(add(futurevalue( 1, 1, n),k=0..n),n=- 1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008 %p A001787 with(finance):seq(add(futurevalue( 2, 1, n),k=0..n)/2,n=-1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008 %t A001787 Table[Sum[Binomial[n, i] i, {i, 0, n}], {n, 0, 30}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 18 2009] %o A001787 (PARI) a(n)=if(n<0,0,n*2^(n-1)) %o A001787 (Other) sage: [lucas_number1(n,4,4) for n in xrange(0, 30)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009] %Y A001787 Partial sums of A001792. Cf. A053109, A001788, A001789. A058922(n+1) = 4*A001787(n). %Y A001787 Row sums of triangle in A003506. Equals A090802(n, 1). %Y A001787 Cf. A000337, A130300, A134083, A002064. %Y A001787 Three other versions, essentially identical, are A085750, A097067, A118442. %Y A001787 Cf. A027471. %Y A001787 Cf. A003945, A059670. %Y A001787 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start) %Y A001787 Equals the first left hand column of A167591. %Y A001787 (End) %Y A001787 Sequence in context: A085750 A097067 A139756 this_sequence A118442 A038592 A048776 %Y A001787 Adjacent sequences: A001784 A001785 A001786 this_sequence A001788 A001789 A001790 %K A001787 nonn,easy,nice %O A001787 0,3 %A A001787 N. J. A. Sloane (njas(AT)research.att.com). %E A001787 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009 Search completed in 0.003 seconds