A336588 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(exp(x) - 1)).

1, 2, 9, 67, 725, 10616, 200767, 4740149, 136113217, 4656324934, 186642121061, 8647446227487, 457854954921949, 27435354945248732, 1844986431192663683, 138229607701444447561, 11464234006789370705537, 1046538415206891196153834, 104623195637603009050593697

Offset: 0

Author

Ilya Gutkovskiy, Jul 26 2020

Keywords

• nonn

Crossrefs

Cf. A000110, A002720, A119392, A336589.

Programs

Mathematica

nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!^2
Table[n! Sum[StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]

PARI

a(n) = n! * sum(k=0, n, stirling(n+1,k+1,2) / k!); \\ Michel Marcus, Jul 27 2020

Formula

a(n) = n! * Sum_{k=0..n} Stirling2(n+1,k+1) / k!.