A228837 a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2*k)*k).

1, 1, 2, 5, 38, 597, 14472, 554653, 44421258, 8933194659, 3408672951784, 1984802013951149, 1803179670478111304, 3323206887194925488269, 15156709454119350064982141, 132889643918499982093215167857, 1784438297905511051093397284187186

Offset

0,3

Comments

Equals the antidiagonal sums of triangle A228836.

Links

G. C. Greubel, Table of n, a(n) for n = 0..86

Formula

Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - Vaclav Kotesovec, added Sep 05 2013, simplified Mar 04 2014

Mathematica

Table[Sum[Binomial[(n-k)^2, (n-2*k)*k], {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Sep 05 2013 *)

Prog

(PARI) {a(n)=sum(k=0, n\2, binomial((n-k)^2, (n-2*k)*k))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A228836, A207138.

Cf. variants: A209331, A228833, A123165.

Sequence in context: A221681 A347070 A290711 * A206155 A135378 A077398

Adjacent sequences: A228834 A228835 A228836 * A228838 A228839 A228840

Keyword

nonn

Author

Paul D. Hanna, Sep 05 2013

Status

approved