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%I #22 Jul 20 2023 07:26:08
%S 1,1,2,5,11,143,1847,24127,2101931,96398196,9362963203,3376252046640,
%T 551993132054154,434634824535802596,528116646162507517308,
%U 372831439174848001477184,2029862948426766042724907818
%N Row sums of A123163.
%H Seiichi Manyama, <a href="/A123165/b123165.txt">Table of n, a(n) for n = 0..94</a>
%F 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
%t A123163[n_, k_]= ((n-k)^2)!/((k^2)!(n^2-2*n*k)!);
%t Table[Sum[A123163[n,k], {k,0,n/2}], {n,0,20}]
%t Table[Sum[Binomial[(n-k)^2,k^2], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 04 2014 *)
%o (PARI) {a(n) = sum(k=0, n\2, binomial((n-k)^2, k^2))} \\ _Seiichi Manyama_, Jan 28 2019
%o (Magma) [(&+[Binomial((n-k)^2, k^2): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jul 19 2023
%o (SageMath)
%o def A123165(n): return sum(binomial((n-k)^2, k^2) for k in range(n//2+1))
%o [A123165(n) for n in range(31)] # _G. C. Greubel_, Jul 19 2023
%Y Cf. A123163, A206851.
%K nonn
%O 0,3
%A _Roger L. Bagula_, Oct 02 2006
%E Edited by _N. J. A. Sloane_, Oct 04 2006
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