A393915 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(2*k,k) * k!.

1, 3, 21, 247, 4065, 85491, 2182693, 65442231, 2250974337, 87294730915, 3765990774261, 178812764864823, 9263097025165921, 519758109395523987, 31395814449356220165, 2030900829581042821111, 140048613574000341166593, 10254546115119932443226691, 794473964275576023742250197

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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).

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

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

a(n) = hypergeom([1/2, -n, -n], [1], 4). - Peter Luschny, May 12 2026

Maple

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

Mathematica

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

Crossrefs

Cf. A002893, A336293.

Sequence in context: A365602 A371006 A355092 * A205319 A377790 A355099

Adjacent sequences: A393912 A393913 A393914 * A393917 A393918 A393919

Keyword

nonn

Author

Vaclav Kotesovec, May 12 2026

Status

approved