A228808 a(n) = Sum_{k=0..n} binomial(n*k, k^2).

1, 2, 4, 20, 296, 10067, 927100, 219541877, 110728186648, 137502766579907, 448577320868198789, 3169529341990169816462, 51243646781214826181569316, 2201837465728010770618930322223, 215520476721579201896200887266792583, 45634827026091489574547858030506357191920

Offset: 0

Author

Paul D. Hanna, Sep 04 2013

Keywords

• nonn

Comments

Ignoring initial term, equals the logarithmic derivative of A228809.

Equals row sums of triangle A228832.

Examples

			L.g.f.: L(x) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 +...
where
exp(L(x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 + 158904*x^6 + 31681195*x^7 +...+ A228809(n)*x^n +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..73

Crossrefs

Cf. A228809, A207136, A167009, A228832.

Programs

Mathematica

Table[Sum[Binomial[n*k, k^2],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Sep 06 2013 *)

PARI

a(n)=sum(k=0,n,binomial(n*k,k^2))
for(n=0,20,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013