A070833 a(n) = Sum_{k=0..n} binomial(10*n,10*k).

1, 2, 184758, 60090032, 139541849878, 94278969044262, 126648421364527548, 111019250117021378442, 125257104438594491956518, 121088185204450642433930072, 128442558588779813655233443038, 128767440665677943753184267342902

Offset: 0

Author

Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..332
• Index entries for linear recurrences with constant coefficients, signature (522,587797,-75135226,-392963125,3200000).

Crossrefs

Sum_{k=0..n} binomial(b*n,b*k): A000079 (b=1), A081294 (b=2), A007613 (b=3), A070775 (b=4), A070782 (b=5), A070967 (b=6), A094211 (b=7), A070832 (b=8), A094213 (b=9), this sequence (b=10). Cf. A000032.

Programs

PARI

a(n) = sum(k=0, n, binomial(10*n, 10*k)); \\ Michel Marcus, Mar 15 2019

PARI

Vec((1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)) + O(x^15)) \\ Colin Barker, Mar 15 2019

Formula

a(n) = 1/10*1024^n+1/5*(-625/2+275/2*sqrt(5))^n+1/5*(-625/2-275/2*sqrt(5))^n+1/5*(123/2+55/2*sqrt(5))^n+1/5*(123/2-55/2*sqrt(5))^n.

G.f.: (1 - 520*x - 404083*x^2 + 37605988*x^3 + 117888625*x^4 - 320000*x^5) / ((1 - 1024*x)*(1 - 123*x + x^2)*(1 + 625*x + 3125*x^2)). - Colin Barker, Mar 15 2019

a(2*n) = (2^(20*n-1) + Lucas(20*n) + 5^(5*n)*Lucas(10*n))/5, for n>0 and for Lucas(n) = A000032(n). - Greg Dresden, Feb 04 2023