A123165 Row sums of A123163.

1, 1, 2, 5, 11, 143, 1847, 24127, 2101931, 96398196, 9362963203, 3376252046640, 551993132054154, 434634824535802596, 528116646162507517308, 372831439174848001477184, 2029862948426766042724907818

Offset: 0

Author

Roger L. Bagula, Oct 02 2006

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..94

Crossrefs

Cf. A123163, A206851.

Programs

Magma

[(&+[Binomial((n-k)^2, k^2): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 19 2023

Mathematica

A123163[n_, k_]= ((n-k)^2)!/((k^2)!(n^2-2*n*k)!);
Table[Sum[A123163[n,k], {k,0,n/2}], {n,0,20}]
Table[Sum[Binomial[(n-k)^2,k^2], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 04 2014 *)

PARI

{a(n) = sum(k=0, n\2, binomial((n-k)^2, k^2))} \\ Seiichi Manyama, Jan 28 2019

SageMath

def A123165(n): return sum(binomial((n-k)^2, k^2) for k in range(n//2+1))
[A123165(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

Formula

Limit_{n-> oo} a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166..., where r = A323778 = 0.365498498219858044579736... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 04 2014

Extensions

Edited by N. J. A. Sloane, Oct 04 2006