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

1, 344, 22296, 615000, 9876000, 108487128, 897376152, 5950405848, 33031486875, 158406862000, 671944398512, 2567519091888, 8965083682032, 28938181326000, 87168786702000, 246953567853744, 662331582918141, 1691011474896264, 4129363811437000, 9684000822437000

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for linear recurrences with constant coefficients, signature (20, -190, 1140, -4845, 15504, -38760, 77520, -125970, 167960, -184756, 167960, -125970, 77520, -38760, 15504, -4845, 1140, -190, 20, -1).

Formula

-(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

From Yahia Kahloune, Dec 23 2013; (Start)

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;

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).

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

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

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

Example

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.

Maple

A086028 := proc(n)
    add( binomial(i+5, 6)^3, i=1..n) ;
end proc:
seq(A086028(n), n=1..30) ; # R. J. Mathar, Dec 22 2013

Mathematica

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

Prog

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

Crossrefs

Cf. A087127, A024166, A085438 - A085442, A086020, A086021 - A086030.

Sequence in context: A231028 A185746 A202904 * A220650 A178191 A172934

Adjacent sequences: A086025 A086026 A086027 * A086029 A086030 A086031

Keyword

easy,nonn

Author

André F. Labossière, Jul 11 2003

Extensions

More terms from Colin Barker, May 02 2014

Status

approved