A348315 - OEIS

%I #22 Dec 24 2024 07:33:21

%S 1,1,4,64,4382,1357136,1597653852,8389021518585,164828345435877580,

%T 14256525628649472111712,4602970880920727147946847283,

%U 6484132480933772335644792339409450,34112054985056318746734374876035089268523

%N a(n) = Sum_{k=0..n} binomial(n^2 - k,n*k).

%H Seiichi Manyama, <a href="/A348315/b348315.txt">Table of n, a(n) for n = 0..57</a>

%F a(n) = A306680(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^(n+1)).

%F a(n) ~ c * 2^(1/2 - n/2 + n^2) / (sqrt(Pi)*exp(1/8)*n), where c = Sum_{m = -oo..+oo} 1/(2^m * exp(m*(2*m+1))) = 1.77058122254033174512511... if n is even and c = Sum_{m = -oo..+oo} 1/(2^(m + 1/2) * exp((m+1)*(2*m+1))) = 1.81629595919505881855931... if n is odd. - _Vaclav Kotesovec_, Oct 12 2021

%t a[n_] := Sum[Binomial[n^2 - k, n*k], {k, 0, n}]; Array[a, 13, 0] (* _Amiram Eldar_, Oct 12 2021 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n^2-k, n*k));

%o (PARI) a(n) = polcoef((1-x)^(n-1)/((1-x)^n-x^(n+1)+x*O(x^n^2)), n^2);

%Y Cf. A167009, A238696, A306680, A348322.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 11 2021