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Stirling transform of A002457.
2

%I #10 Jun 23 2023 17:51:24

%S 1,6,36,236,1686,13028,108078,956348,8976708,88962160,927129786,

%T 10125636716,115543526476,1373933166848,16985192456410,

%U 217851008508220,2893517713599370,39732264695056772,563187218351672330,8229159647194683140,123795221970087313340

%N Stirling transform of A002457.

%H Alois P. Heinz, <a href="/A066053/b066053.txt">Table of n, a(n) for n = 0..538</a>

%F a(n) = sum(stirling2(n, k)*(2*k + 2)!/(2*k!*(k + 1)!), k = 0..n), n = 0, 1, ...;

%F E.g.f.: exp(2*exp(x) - 2)*(BesselI(0, 2*exp(x) - 2) + 4*BesselI(0, 2*exp(x) - 2)*(exp(x) - 1) + 4*(exp(x) - 1)*BesselI(1, 2*exp(x) - 2)).

%p b:= proc(n, m) option remember; `if`(n=0,

%p (2*m+1)!/m!^2, m*b(n-1, m)+b(n-1, m+1))

%p end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Jun 23 2023

%t Table[Sum[StirlingS2[n, k]*(2*k+2)!/(2*k!*(k+1)!), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 01 2018 *)

%Y Cf. A002457, A064856.

%K nonn

%O 0,2

%A _Karol A. Penson_, Nov 30 2001