A349513 a(n) = n! * Sum_{k=0..n} (2*k)! / (k!)^3.

1, 3, 12, 56, 294, 1722, 11256, 82224, 670662, 6084578, 61030536, 672041328, 8067200092, 104884001796, 1468416141744, 22026397243680, 352422956979270, 5991192602253810, 107841475915703880, 2048988077743637520, 40979761692719279220, 860574996085362738060

Offset: 0

Author

Ilya Gutkovskiy, Nov 20 2021

Keywords

• nonn

Crossrefs

Cf. A000142, A000522, A000984, A002426, A026375, A052143, A336293.

Programs

Mathematica

Table[n! Sum[(2 k)!/(k!)^3, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[Exp[2 x] BesselI[0, 2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

PARI

a(n) = n! * sum(k=0, n, (2*k)! / (k!)^3) \\ Andrew Howroyd, Nov 20 2021

Formula

E.g.f.: exp(2*x) * BesselI(0,2*x) / (1 - x).

a(n) = Sum_{k=0..n} binomial(n,k) * A000984(k) * A000142(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A002426(k) * A000522(n-k).

a(n) ~ exp(2) * BesselI(0,2) * n!. - Vaclav Kotesovec, Nov 20 2021