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%I #13 Mar 02 2014 08:39:40
%S 1,1,3,13,72,481,3745,33209,329868,3624270,43608474,570008803,
%T 8039735704,121673027607,1966231022067,33786076421499,615043147866660,
%U 11822938288619344,239298079351004608,5086498410027323134,113278368771499790136,2637549737582063583274,64082443707327038140602,1621782672366231029685407
%N Stirling transform of Catalan numbers: Sum_{k=0..n} |Stirling1(n,k)|*C(2*k,k)/(k+1).
%H Vaclav Kotesovec, <a href="/A086662/b086662.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: hypergeom([1/2], [2], -4*log(1-x)) = 1/(1-x)^2*(BesselI(0, 2*log(1-x))+BesselI(1, 2*log(1-x))).
%F a(n)=(1/(2*pi))*int(product(x+k,k,0,n-1)*sqrt((4-x)/x),x,0,4) (moment representation). [_Paul Barry_, Jul 26 2010]
%t CoefficientList[Series[(BesselI[0, 2*Log[1-x]] + BesselI[1, 2*Log[1-x]]) / (1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Mar 02 2014 *)
%t Table[Sum[Abs[StirlingS1[n,k]]*Binomial[2*k,k]/(k+1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 02 2014 *)
%o (PARI) a(n)=sum(k=0,n, abs(stirling(n,k,1)) * binomial(2*k,k)/(k+1) ); \\ _Joerg Arndt_, Mar 02 2014
%Y Cf. A000108, A008275, A064856.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Sep 12 2003