A228833 a(n) = Sum_{k=0..[n/2]} binomial((n-k)*k, k^2).

1, 1, 2, 3, 5, 20, 77, 437, 5509, 54475, 1031232, 31874836, 789351469, 47552777430, 3302430043985, 223753995897916, 39177880844093733, 5954060239110086680, 1226026438114057710320, 551315671593483499670137, 188615011023291125237647365, 124995445742889226418307452940

Offset: 0

Author

Paul D. Hanna, Sep 04 2013

Keywords

• nonn

Comments

Equals antidiagonal sums of triangle A228832.

Crossrefs

Cf. A228832.

Programs

Mathematica

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

PARI

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

Formula

Limit n->infinity a(n)^(1/n^2) = ((1-r)/(1-2*r))^(r/2) = 1.171233876693210503..., where r = A323773 = 0.366320150305283... is the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r). - Vaclav Kotesovec, Sep 06 2013