A346224 a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} 1 / ((n-2*k)! * 4^k * k!).

1, 1, 3, 15, 114, 1170, 15570, 256410, 5103000, 119773080, 3264445800, 101784097800, 3591396824400, 141958074258000, 6236035482877200, 302218901402418000, 16060366291617648000, 930654556409161584000, 58524794739862410960000, 3976525824684785163792000

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 ).

a(n) = n! * Sum_{k=0..n} Stirling1(n,k) * Bell(k) / 2^(n-k).

D-finite with recurrence a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2.

a(n) ~ sqrt(Pi) * n^((3*n + 1)/2) / (2^(n/2) * exp((3*n + 1)/2 - sqrt(2*n))). - Vaclav Kotesovec, Jul 17 2021

Mathematica

Table[(n!)^2 Sum[1/((n - 2 k)! 4^k k!), {k, 0, Floor[n/2]}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x + x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Prog

(PARI) a(n) = (n!)^2 * sum(k=0, n\2, 1/((n-2*k)!*4^k*k!)); \\ Michel Marcus, Jul 17 2021

Crossrefs

Cf. A000085, A000898, A023998, A080599, A239840.

Sequence in context: A357794 A335531 A166885 * A082163 A331122 A342167

Adjacent sequences: A346221 A346222 A346223 * A346225 A346226 A346227

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jul 16 2021

Status

approved