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

%I A305406 #8 Feb 16 2025 08:33:54
%S A305406 1,2,8,40,234,1544,11242,89016,758504,6900012,66590782,678322704,
%T A305406 7262393832,81431657220,953339019606,11622207372104,147199295291518,
%U A305406 1932876310310488,26265519359529974,368752956750812256,5340795881536757632,79691179458925839676,1223524383429928039306
%N A305406 Expansion of Sum_{k>=0} binomial(2*k,k)*x^k/Product_{j=1..k} (1 - j*x).
%C A305406 Stirling transform of A000984.
%H A305406 Alois P. Heinz, <a href="/A305406/b305406.txt">Table of n, a(n) for n = 0..539</a>
%H A305406 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A305406 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A305406 E.g.f.: exp(2*(exp(x) - 1))*BesselI(0,2*(exp(x) - 1)).
%F A305406 a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(2*k,k).
%p A305406 b:= proc(n, m) option remember;
%p A305406      `if`(n=0, binomial(2*m, m), m*b(n-1, m)+b(n-1, m+1))
%p A305406     end:
%p A305406 a:= n-> b(n, 0):
%p A305406 seq(a(n), n=0..23);  # _Alois P. Heinz_, Aug 04 2021
%t A305406 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]
%t A305406 nmax = 22; CoefficientList[Series[Exp[2 (Exp[x] - 1)] BesselI[0, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
%t A305406 Table[Sum[StirlingS2[n, k] Binomial[2 k, k], {k, 0, n}], {n, 0, 22}]
%Y A305406 Cf. A000984, A064856.
%K A305406 nonn
%O A305406 0,2
%A A305406 _Ilya Gutkovskiy_, May 31 2018