A123165 Row sums of A123163.

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

Offset

0,3

Links

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

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

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 *)

Prog

(PARI) {a(n) = sum(k=0, n\2, binomial((n-k)^2, k^2))} \\ Seiichi Manyama, Jan 28 2019
(Magma) [(&+[Binomial((n-k)^2, k^2): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 19 2023
(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

Crossrefs

Cf. A123163, A206851.

Sequence in context: A283300 A069506 A239900 * A098438 A225955 A236073

Adjacent sequences: A123162 A123163 A123164 * A123166 A123167 A123168

Keyword

nonn

Author

Roger L. Bagula, Oct 02 2006

Extensions

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

Status

approved