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a(n) = Sum_{i=1..n} C(i+5,6)^3.
20

%I #31 Jun 11 2023 12:11:53

%S 1,344,22296,615000,9876000,108487128,897376152,5950405848,

%T 33031486875,158406862000,671944398512,2567519091888,8965083682032,

%U 28938181326000,87168786702000,246953567853744,662331582918141,1691011474896264,4129363811437000,9684000822437000

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

%H G. C. Greubel, <a href="/A086028/b086028.txt">Table of n, a(n) for n = 1..5000</a>

%H <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 -(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 From _Yahia Kahloune_, Dec 23 2013; (Start)

%F 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 where F6(n) is Sum_(i=1..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 The values of F6(n), (n=0...9) are: 1, 1716, 10725, 39754, 112827, 270348, 575107, 1119210, 2031933, 3488500, .... (End)

%F 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 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 a(4) = Sum_(i=1..4)C(5+i,6)^3 = C(10,7)*[-15*112827 + 63063*(7*C(15,12) + 195*C(14,12) + 920*C(13,12) + 920*C(12,12)]/415701 = 615000.

%p A086028 := proc(n)

%p add( binomial(i+5,6)^3,i=1..n) ;

%p end proc:

%p seq(A086028(n),n=1..30) ; # _R. J. Mathar_, Dec 22 2013

%t Table[Sum[Binomial[k+5,6]^3, {k,1,n}], {n,1,30}] (* _G. C. Greubel_, Nov 22 2017 *)

%o (PARI) for(n=1, 30, print1(sum(k=1,n, binomial(k+5,6)^3), ", ")) \\ _G. C. Greubel_, Nov 22 2017

%o (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 Cf. A087127, A024166, A085438 - A085442, A086020, A086021 - A086030.

%K easy,nonn

%O 1,2

%A _André F. Labossière_, Jul 11 2003

%E More terms from _Colin Barker_, May 02 2014