A064856 Stirling transform of Catalan numbers: a(n) = Sum_{k=0..n} stirling2(n,k)*binomial(2*k,k)/(k+1).
1, 1, 3, 12, 59, 338, 2185, 15613, 121553, 1020170, 9154963, 87276995, 879242215, 9319182044, 103537712361, 1201967382478, 14540040004755, 182840037042560, 2384985091689409, 32209645344213417, 449608555748234353, 6476887237235672388, 96156363230696213447
Offset: 0
Links
- Robert Israel, Table of n, a(n) for n = 0..400
Programs
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Maple
seq(add(Stirling2(n,k)*binomial(2*k,k)/(k+1),k=0..n), n=0..50); # Robert Israel, Sep 16 2016
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Mathematica
Table[Sum[StirlingS2[n,k] Binomial[2k,k]/(k+1),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 01 2011 *) -
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 */
Formula
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
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...
E.g.f.: hypergeom([1/2], [2], 4*(exp(x)-1)). - Vladeta Jovovic, Sep 11 2003