A178824 a(n) = Sum_{k=0..n} binomial(n,k)^4/(n+1).

1, 1, 6, 41, 362, 3542, 37692, 424377, 4990722, 60704138, 758665388, 9694652838, 126203947828, 1668947978908, 22370427181624, 303383342784729, 4156846359584754, 57473870722327874, 801081711581734764, 11246487794657694810, 158920231643036635860, 2258896576436091238860

Offset: 0

Author

Paul D. Hanna, Dec 27 2010

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..830

Crossrefs

Cf. A000108, A005260, A131428.

Programs

GAP

List([0..20], n-> Sum([0..n], k-> Binomial(n,k)^4/(n+1) )); # G. C. Greubel, Jan 22 2019

Magma

[(&+[Binomial(n,k)^4/(n+1): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 22 2019

Maple

a:=n->add(binomial(n,k)^4/(n+1),k=0..n): seq(a(n),n=0..20); # Muniru A Asiru, Jan 22 2019

Mathematica

Table[Sum[Binomial[n,k]^4/(n+1), {k,0,n}], {n,0,20}] (* G. C. Greubel, Jan 22 2019 *)

PARI

{a(n)=sum(k=0, n, binomial(n, k)^4)/(n+1)}

Sage

[sum(binomial(n,k)^4/(n+1) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jan 22 2019

Formula

a(n) = A005260(n)/(n+1).

a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^2. - Seiichi Manyama, Mar 26 2025

a(n) ~ 2^(4*n + 1/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 26 2025