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
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Oct 02 2006
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2006
STATUS
approved