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A323632 Stirling transform of Jacobsthal numbers (A001045).

%I A323632 #13 Aug 06 2021 08:40:39
%S A323632 0,1,2,7,31,152,813,4741,29956,203305,1470795,11276718,91221419,
%T A323632 775677177,6910797962,64326920851,623981351195,6293426736344,
%U A323632 65867162316433,714062197266081,8005397253530924,92676194887133693,1106385117766336919,13603803900252612966,172082332173918135687
%N A323632 Stirling transform of Jacobsthal numbers (A001045).
%H A323632 Alois P. Heinz, <a href="/A323632/b323632.txt">Table of n, a(n) for n = 0..558</a>
%F A323632 E.g.f.: (exp(2*(exp(x) - 1)) - exp(1 - exp(x)))/3.
%F A323632 a(n) = Sum_{k=0..n} Stirling2(n,k)*A001045(k).
%F A323632 a(n) = (A001861(n) - A000587(n))/3.
%p A323632 b:= proc(n, m) option remember;
%p A323632      `if`(n=0, round(2^m/3), m*b(n-1, m)+b(n-1, m+1))
%p A323632     end:
%p A323632 a:= n-> b(n, 0):
%p A323632 seq(a(n), n=0..24);  # _Alois P. Heinz_, Aug 06 2021
%t A323632 nmax = 24; CoefficientList[Series[(Exp[2 (Exp[x] - 1)] - Exp[1 - Exp[x]])/3, {x, 0, nmax}], x] Range[0, nmax]!
%t A323632 Table[Sum[StirlingS2[n, k] (2^k - (-1)^k)/3, {k, 0, n}], {n, 0, 24}]
%t A323632 Table[(BellB[n, 2] - BellB[n, -1])/3, {n, 0, 24}]
%Y A323632 Cf. A000587, A001045, A001861, A263575, A263576.
%K A323632 nonn
%O A323632 0,3
%A A323632 _Ilya Gutkovskiy_, Jan 21 2019