A305406 Expansion of Sum_{k>=0} binomial(2*k,k)*x^k/Product_{j=1..k} (1 - j*x).

1, 2, 8, 40, 234, 1544, 11242, 89016, 758504, 6900012, 66590782, 678322704, 7262393832, 81431657220, 953339019606, 11622207372104, 147199295291518, 1932876310310488, 26265519359529974, 368752956750812256, 5340795881536757632, 79691179458925839676, 1223524383429928039306

Offset: 0

Author

Ilya Gutkovskiy, May 31 2018

Keywords

• nonn

Comments

Stirling transform of A000984.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..539
• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A000984, A064856.

Programs

Maple

b:= proc(n, m) option remember;
     `if`(n=0, binomial(2*m, m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

Mathematica

nmax = 22; CoefficientList[Series[Sum[Binomial[2 k, k] x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[2 (Exp[x] - 1)] BesselI[0, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Binomial[2 k, k], {k, 0, n}], {n, 0, 22}]

Formula

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

a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(2*k,k).