cp's OEIS Frontend

A086028 a(n) = Sum_{i=1..n} C(i+5,6)^3.

%I A086028 #40 Jan 14 2025 11:06:37
%S A086028 1,344,22296,615000,9876000,108487128,897376152,5950405848,
%T A086028 33031486875,158406862000,671944398512,2567519091888,8965083682032,
%U A086028 28938181326000,87168786702000,246953567853744,662331582918141,1691011474896264,4129363811437000,9684000822437000
%N A086028 a(n) = Sum_{i=1..n} C(i+5,6)^3.
%H A086028 G. C. Greubel, <a href="/A086028/b086028.txt">Table of n, a(n) for n = 1..5000</a>
%H A086028 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 13.
%H A086028 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).
%F A086028 -(n-1)^3*a(n) +2*(n+2)*(n^2 +4*n +31)*a(n-1) -(n+5)^3*a(n-2)=0. - _R. J. Mathar_, Dec 22 2013
%F A086028 From _Yahia Kahloune_, Dec 23 2013; (Start)
%F A086028 a(n) = C(n+6,7)*(-15*F6(n) + 63063*(7*C(n+11,12) + 195*C(n+10,12) + 920*C(n+9,12) + 920*C(n+8,12) + 195*C(n+7,12) + 7*C(n+6,12)))/415701;
%F A086028 where F6(n) = Sum_{i=0..6} (-1)^i*C(6+i,i)*C(n+6,i) = C(6,0)*C(n+6,0) - C(7,1)*C(n+6,1) + C(8,2)*C(n+6,2) - C(9,3)*C(n+6,3) + C(10,4)*C(n+6,4) - C(11,5)*C(n+6,5) + C(12,6)*C(n+6,6).
%F A086028 The values of F6(n), (n=0...9) are: 1, 1716, 10725, 39754, 112827, 270348, 575107, 1119210, 2031933, 3488500, .... (End)
%F A086028 G.f.: x*(x^12 +324*x^11 +15606*x^10 +233300*x^9 +1424925*x^8 +4050864*x^7 +5703096*x^6 +4050864*x^5 +1424925*x^4 +233300*x^3 +15606*x^2 +324*x +1) / (x -1)^20. - _Colin Barker_, May 02 2014
%F A086028 a(n) = (n/120679663104000)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-864000 + 2116800*n + 772737840*n^2 + 3398930472*n^3 + 6406454992 *n^4 + 6701566410*n^5 + 4302755765*n^6 + 1780394616*n^7 + 484074591*n^8 + 85975890*n^9 + 9604595*n^10 + 612612*n^11 + 17017*n^12). - _G. C. Greubel_, Nov 22 2017
%e A086028 a(4) = Sum_{i=1..4} C(i+5,6)^3 = C(6,6)^3 + C(7,6)^3 + C(8,6)^3 + C(9,6)^3 = 1^3 + 7^3 + 28^3 + 84^3 = 615000.
%p A086028 A086028 := proc(n)
%p A086028     add( binomial(i+5,6)^3,i=1..n) ;
%p A086028 end proc:
%p A086028 seq(A086028(n),n=1..30) ; # _R. J. Mathar_, Dec 22 2013
%t A086028 Table[Sum[Binomial[k+5,6]^3, {k,1,n}], {n,1,30}] (* _G. C. Greubel_, Nov 22 2017 *)
%o A086028 (PARI) for(n=1, 30, print1(sum(k=1,n, binomial(k+5,6)^3), ", ")) \\ _G. C. Greubel_, Nov 22 2017
%o A086028 (Magma) [(n/120679663104000)*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-864000 + 2116800*n + 772737840*n^2 + 3398930472*n^3 + 6406454992 *n^4 + 6701566410*n^5 + 4302755765*n^6 + 1780394616*n^7 + 484074591*n^8 + 85975890*n^9 + 9604595*n^10 + 612612*n^11 + 17017*n^12): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y A086028 Cf. A087127, A024166, A085438 - A085442, A086020, A086021 - A086030.
%K A086028 easy,nonn
%O A086028 1,2
%A A086028 _André F. Labossière_, Jul 11 2003
%E A086028 More terms from _Colin Barker_, May 02 2014