%I #93 Aug 08 2024 23:07:38
%S 1,8,35,112,294,672,1386,2640,4719,8008,13013,20384,30940,45696,65892,
%T 93024,128877,175560,235543,311696,407330,526240,672750,851760,
%U 1068795,1330056,1642473,2013760,2452472,2968064,3570952,4272576,5085465
%N a(n) = binomial(n+5,5)*(n+3)/3.
%C Sequence is n^2*(n^2-1)*(n^2-4)/360 if offset 3.
%C If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-7) is the number of 7-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%C 6-dimensional square numbers, fifth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+5,i+5)*b(i), where b(i) = [1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%C Sequence of the absolute values of the z^2 coefficients divided by 5 of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - _Johannes W. Meijer_, Mar 07 2009
%C 2*a(n) is number of ways to place 5 queens on an (n+5) X (n+5) chessboard so that they diagonally attack each other exactly 10 times. The maximal possible attack number, p=binomial(k,2)=10 for k=5 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - _Antal Pinter_, Dec 27 2015
%C Ehrhart polynomial for the Chan-Robbins-Yuen polytope CRY_4. [De Loera et al.] - _N. J. A. Sloane_, Apr 16 2016
%C Coefficients in the terminating series identity 1 - 8*n/(n + 7) + 35*n*(n - 1)/((n + 7)*(n + 8)) - 112*n*(n - 1)*(n - 2)/((n + 7)*(n + 8)*(n + 9)) + ... = 0 for n = 1,2,3,.... Cf. A005585 and A050486. - _Peter Bala_, Feb 18 2019
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.
%H Vincenzo Librandi, <a href="/A040977/b040977.txt">Table of n, a(n) for n = 0..1000</a>
%H Jesús A. De Loera, Fu Liu and Ruriko Yoshida, <a href="https://www.emis.de/journals/JACO/Volume30_1/m6627810x2013373.html">A generating function for all semi-magic squares and the volume of the Birkhoff polytope</a>, J. Algebraic Combin., Vol. 30, No. 1 (2009), pp. 113-139. See page 138, n=4 entry in table.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = (-1)^n*A053120(2*n+6, 6)/32, (1/32 of seventh unsigned column of Chebyshev T-triangle, zeros omitted).
%F G.f.: (1+x)/(1-x)^7.
%F a(n-3) = Sum_{i+j+k=n} i*j*k^2. - _Benoit Cloitre_, Nov 01 2002
%F a(n) = 2*binomial(n+6, 6) - binomial(n+5, 5). - _Paul Barry_, Mar 04 2003
%F a(n-3) = 1/(1!*2!*3!)*Sum_{1 <= x_1, x_2, x_3 <= n} |det V(x_1,x_2,x_3)| = 1/12*Sum_{1 <= i,j,k <= n} |(i-j)(i-k)(j-k)|, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - _Peter Bala_, Sep 13 2007
%F a(n) = binomial(n+5,5) + 2*binomial(n+5,6). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F a(n) = (n+1)*(n+2)*(n+3)^2*(n+4)*(n+5)/360. - _Wesley Ivan Hurt_, May 05 2015
%F a(n) = A000579(n+5) + A000579(n+6). - _R. J. Mathar_, Nov 29 2015
%F Sum_{n>=0} 1/a(n) = 15*Pi^2 - 1175/8. - _Jaume Oliver Lafont_, Jul 11 2017
%F Sum_{n>=0} (-1)^n/a(n) = 15*Pi^2/2 - 585/8. - _Amiram Eldar_, Jan 24 2022
%p with(combinat); A040977 := n->binomial(n+5,5)*(n+3)/3;
%p a:=n->(sum((numbcomp(n,6)), j=4..n))/3:seq(a(n), n=6..38); # _Zerinvary Lajos_, Aug 26 2008
%p nmax:=34; for n from 0 to nmax do fz(n):=product((1-m*z)^(n+1-m),m=1..n); c(n):= abs(coeff(fz(n),z,2))/5; end do: a:=n-> c(n): seq(a(n), n=2..nmax); # _Johannes W. Meijer_, Mar 07 2009
%t CoefficientList[Series[(1 + x) / (1 - x)^7, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,8,35,112,294,672,1386},40] (* _Harvey P. Dale_, Feb 20 2016 *)
%o (Magma) [Binomial(n+5, 5) + 2*Binomial(n+5, 6): n in [0..35]]; // _Vincenzo Librandi_, Jun 09 2013
%o (PARI) vector(20,n,n--;2*binomial(n+6,6)-binomial(n+5,5)) \\ _Derek Orr_, May 05 2015
%o (PARI) Vec((1+x)/(1-x)^7 + O(x^100)) \\ _Altug Alkan_, Nov 29 2015
%Y Partial sums of A005585.
%Y Cf. A000292, A000579, A050486, A053120, A133111, A133112.
%Y Cf. A156925, A157703. - _Johannes W. Meijer_, Mar 07 2009
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Dec 14 1999