A393915 - OEIS

%I #39 May 12 2026 09:50:37

%S 1,3,21,247,4065,85491,2182693,65442231,2250974337,87294730915,

%T 3765990774261,178812764864823,9263097025165921,519758109395523987,

%U 31395814449356220165,2030900829581042821111,140048613574000341166593,10254546115119932443226691,794473964275576023742250197

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

%F Recurrence: n*(2*n - 5)*a(n) = (16*n^3 - 52*n^2 + 38*n - 11)*a(n-1) - (n-1)*(32*n^3 - 128*n^2 + 162*n - 71)*a(n-2) + 16*(n-2)^2*(n-1)*(2*n - 3)*a(n-3).

%F a(n) ~ Pi^(-1/2) * 2^(2*n) * exp(sqrt(n) - n - 1/8) * n^(n - 1/4) * (1 + 61/(96*sqrt(n))).

%F Sum_{n>=0} a(n)*x^n/(n!)^2 = B(x)*C(x) where B(x) = Sum_{n>=0} x^n/(n!)^2 and C(x) = Sum_{n>=0} binomial(2*n,n)*x^n/n!. - _Paul D. Hanna_, May 12 2026

%F a(n) = hypergeom([1/2, -n, -n], [1], 4). - _Peter Luschny_, May 12 2026

%p a := n -> hypergeom([1/2, -n, -n], [1], 4): seq(simplify(a(n)), n = 0..18);  # _Peter Luschny_, May 12 2026

%t Table[Sum[Binomial[n, k]^2 * Binomial[2*k, k] * k!, {k, 0, n}], {n, 0, 20}]

%Y Cf. A002893, A336293.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, May 12 2026