Search: id:A002415 Results 1-1 of 1 results found. %I A002415 M4135 N1714 %S A002415 0,0,1,6,20,50,105,196,336,540,825,1210,1716,2366,3185,4200,5440,6936, %T A002415 8721,10830,13300,16170,19481,23276,27600,32500,38025,44226,51156, %U A002415 58870,67425,76880,87296,98736,111265,124950,139860,156066,173641 %N A002415 4-dimensional pyramidal numbers: n^2*(n^2-1)/12. %C A002415 Also number of ways to legally insert two pairs of parentheses into a string of m := n-1 letters. (There are initially 2C(m+4,4) (A034827) ways to insert the parentheses, but we must subtract 2(m+1) for illegal clumps of 4 parentheses, 2m(m+1) for clumps of 3 parentheses, C(m+1, 2) for 2 clumps of 2 parentheses and (m-1)C(m+1,2) for 1 clump of 2 parentheses, giving m(m+1)^2(m+2)/12 = n^2*(n^2-1)/12.) See also A000217. %C A002415 E.g. for n=2 there are 6 ways: ((a))b, ((a)b), ((ab)), (a)(b), (a(b)), a((b)). %C A002415 Let M_n denotes the n X n matrix M_n(i,j)=(i+j); then the characteristic polynomial of M_n is x^(n-2) * (x^2-A002378(n)*x - a(n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 09 2002 %C A002415 Let M_n denotes the n X n matrix M_n(i,j)=(i-j); then the characteristic polynomial of M_n is x^n + a(n)x^(n-2). - Michael Somos, Nov 14 2002 %C A002415 a(n)+1 is the determinant of the n X n matrix M with M(i,i)=1, M(i,j)=i-j. - Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003 %C A002415 Number of permutations of [n] which avoid the pattern 132 and have exactly 2 descents. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004 %C A002415 Number of tilings of a <2,n,2> hexagon. %C A002415 a(n) = number of squares with corners on an n X n grid. See also A024206, A108279. %C A002415 Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005 %C A002415 Number of distinct components of the Riemann curvature tensor. - Gene Ward Smith (genewardsmith(AT)gmail.com), Apr 24 2006 %C A002415 a(n) is the number of 4 X 4 matrices (symmetrical about each diagonal) M = [a,b,c,d;b,e,f,c;c,f,e,b;d,c,b,a] with a+b+c+d=b+e+f+c=n+2; (a, b,c,d,e,f natural numbers). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 11 2007 %C A002415 If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 5-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Sep 19 2007 %C A002415 a(n) = number of Dyck (n+1)-paths with exactly n-1 peaks. - David Callan (callan(AT)stat.wisc.edu), Sep 20 2007 %C A002415 Starting (1,6,20,50,...) = third partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+3,i+3)*b(i)}, where b(i)=[1,2, 0,0,0,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009] %C A002415 4-dimensional square numbers. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009] %D A002415 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A002415 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A002415 O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. %D A002415 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195. %D A002415 S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.165). %D A002415 R. Euler and J. Sadek, "The Number of Squares on a Geoboard", Journal of Recreational Mathematics, 251-5 30(4) 1999-2000 Baywood Pub. NY %D A002415 Franz, Reinhard O. W. and Earnshaw, Berton A. A constructive enumeration of meanders. Ann. Comb. 6 (2002), no. 1, 7-17. %D A002415 G. Kreweras, Traitemant simultane du "Probleme de Young" et du "Probleme de Simon Newcomb", Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle. Institut de Statistique, Universit\'{e} de Paris, 10 (1967), 23-31. %D A002415 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238. %H A002415 T. D. Noe, Table of n, a(n) for n=0..1000 %H A002415 Index entries for sequences related to linear recurrences with constant coefficients %H A002415 Milan Janjic, Two Enumerative Functions %H A002415 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 A002415 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A002415 H. Bottomley, Illustration of initial terms %H A002415 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A002415 Index entries for sequences related to Chebyshev polynomials. %F A002415 G.f.: x^2*(1+x)/(1-x)^5. %F A002415 a(n) = sum(i = 0 to n) [(n-i)*i^2] = a(n-1)+A000330(n-1) = A000217(n)*A000292(n-2)/ n = A000217(n)*A000217(n-1)/3 = A006011(n-1)/3 - Henry Bottomley (se16(AT)btinternet.com), Oct 19 2000 %F A002415 a(n)=2*C(n+2, 4)-C(n+1, 3). - Paul Barry (pbarry(AT)wit.ie), Mar 04 2003 %F A002415 a(n)=C(n+2, 4)+C(n+3, 4). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003 %F A002415 A002415[n-1]=C[n+3, 5]-(C[n, 5]-C[n, 4]-2*C[n, 3]-C[n, 2]). - Labos E. (labos(AT)ana.sote.hu), Apr 30 2003 %F A002415 a(n)=sum(k=1, n, sum(i=1, k-1, i^2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 15 2003 %F A002415 Convolution of natural numbers (A001477) with squares (A000290) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 06 2006 %F A002415 a(n) = n C(n+1, 3)/2 = C(n+1, 3)C(n+1,2)/(n+1) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006 %F A002415 a(n) = A006011(n)/3 = A008911(n)/2 = A047928(n-1)/12 = A083374(n-1)/6. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007 %F A002415 a(n) = 1/2*sum {1 <= x_1, x_2 <= n} (det V(x_1,x_2))^2 = 1/2*sum {1 <= i,j <= n} (i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala (pbala(AT)toucansurf.com), Sep 21 2007 %F A002415 a(n)=C(n^2,2)/6,n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008 %F A002415 Starting n (-2,-1,0,1,2,...) a(n)=C(n+3,3)+2*C(n+3,4) [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009] %p A002415 a:=n->sum(sum(n^2/12, j=2..n),k=0..n): seq(a(n), n=0..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007 %p A002415 A002415:=-(1+z)/(z-1)**5; [S. Plouffe in his 1992 dissertation. Gives sequence without initial zeros.] %p A002415 seq(binomial(n^2,2)/6,n=0..38); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 07 2008 %p A002415 a:=n->(sum((numbperm(n,3)), j=2..n)):seq(a(n)/12, n=1..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2008 %p A002415 a:=n->sum((n-j)^2*j,j=0..n): seq(a(n), n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008 %p A002415 a:=n->(sum((numbcomp(n,4)), j=3..n))/2:seq(a(n), n=2..40); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008] %t A002415 Table[(n^4 -n^2 )/12, {n,0, 40}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007 %t A002415 s1=s2=s3=0;lst={};Do[a=n+(n+1);s1+=a;s2+=s1;s3+=s2;AppendTo[lst,s3],{n, 0,5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 04 2009] %o A002415 (PARI) a(n)=n^2*(n^2-1)/12 %o A002415 (PARI) a(n)=sum(k=1,n,sum(m=1,k,sum(i=1,m,(2*i-1)))) - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 05 2007 %o A002415 sage: [lucas_number1(3,n^2,n^2)/12 for n in xrange(0,39)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008 %Y A002415 a(n)= ((-1)^n)*A053120(2*n, 4)/8 (one eighth of fifth unsigned column of Chebyshev T-triangle, zeros omitted). Cf. A001296. %Y A002415 Second row of array A103905. %Y A002415 Third column of Narayana numbers A001236. %Y A002415 Cf. A006011, A008911, A047928, A083374. %Y A002415 Cf. A006542, A047819, A107891. %Y A002415 Partial sums of A000330. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009] %Y A002415 Sequence in context: A055455 A050768 A063488 this_sequence A052515 A067117 A119365 %Y A002415 Adjacent sequences: A002412 A002413 A002414 this_sequence A002416 A002417 A002418 %K A002415 nonn,easy,nice %O A002415 0,4 %A A002415 N. J. A. Sloane (njas(AT)research.att.com). %E A002415 More terms from Larry Reeves (larryr(AT)acm.org), Oct 19 2000 %E A002415 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009 Search completed in 0.003 seconds