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%I #27 Aug 04 2021 10:08:18
%S 1,1,3,12,59,338,2185,15613,121553,1020170,9154963,87276995,879242215,
%T 9319182044,103537712361,1201967382478,14540040004755,182840037042560,
%U 2384985091689409,32209645344213417,449608555748234353,6476887237235672388,96156363230696213447
%N Stirling transform of Catalan numbers: a(n) = Sum_{k=0..n} stirling2(n,k)*binomial(2*k,k)/(k+1).
%H Robert Israel, <a href="/A064856/b064856.txt">Table of n, a(n) for n = 0..400</a>
%F O.g.f.: Sum_{n>=1} C(2*n,n)/(n+1) * x^n / Product_{k=0..n} (1-k*x). - _Paul D. Hanna_, Jul 20 2011
%F E.g.f.: exp(2*exp(z)-2)*(BesselI(0, 2*exp(z)-2)-BesselI(1, 2*exp(z)-2)). Representation as a sum of an infinite series involving the confluent hypergeometric function 1F1, in Maple notation: a(n)=evalf(sum('k'^n*2^(2*'k')*GAMMA('k'+1/2)*evalf(hypergeom(['k'+1/2], ['k'+2], -4))/(sqrt(Pi)*'k'!*('k'+1)!), 'k'=0..infinity)), n=0, 1...
%F E.g.f.: hypergeom([1/2], [2], 4*(exp(x)-1)). - _Vladeta Jovovic_, Sep 11 2003
%p seq(add(Stirling2(n,k)*binomial(2*k,k)/(k+1),k=0..n), n=0..50); # _Robert Israel_, Sep 16 2016
%t Table[Sum[StirlingS2[n,k] Binomial[2k,k]/(k+1),{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Nov 01 2011 *)
%o (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/(m!*(m+1)!)*x^m/prod(k=1, m, 1-k*x+x*O(x^n))), n)} /* _Paul D. Hanna_, Jul 20 2011 */
%Y Cf. A000108, A066053, A086672.
%K nice,nonn
%O 0,3
%A _Karol A. Penson_, Oct 08 2001