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%I #11 Aug 04 2021 10:11:39
%S 1,1,3,15,127,1895,53071,2953575,337064047,79446381319,38491200186831,
%T 38046637826801703,76226441027901385519,308075833912652114006087,
%U 2503633988838391023366024079,40826169678526460459483237927271,1334110729147927667553970495057395439
%N a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^binomial(k,2).
%C Stirling transform of A006125.
%H Alois P. Heinz, <a href="/A335390/b335390.txt">Table of n, a(n) for n = 0..82</a>
%F G.f.: Sum_{k>=0} 2^binomial(k,2) * x^k / Product_{j=1..k} (1 - j*x).
%F E.g.f.: Sum_{k>=0} 2^binomial(k,2) * (exp(x) - 1)^k / k!.
%F a(n) ~ 2^(n*(n-1)/2). - _Vaclav Kotesovec_, Jun 05 2020
%p a:= n-> add(Stirling2(n, k)*2^(k*(k-1)/2), k=0..n):
%p seq(a(n), n=0..19); # _Alois P. Heinz_, Jun 05 2020
%t Table[Sum[StirlingS2[n, k] 2^Binomial[k, 2], {k, 0, n}], {n, 0, 16}]
%o (PARI) a(n) = sum(k=0, n, stirling(n,k,2) * 2^binomial(k,2)); \\ _Michel Marcus_, Jun 05 2020
%Y Cf. A006024, A006125.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 04 2020
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