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%I #15 Aug 04 2021 09:23:37
%S 2,5,16,60,253,1178,5976,32623,189702,1166720,7554877,51351254,
%T 365560784,2720255911,21121563036,170839106566,1437200307921,
%U 12556366592382,113755900474652,1067028469382353,10346222830388738,103538470949470066,1067747451140472577,11330777204488565252
%N Expansion of e.g.f. Sum_{k>=1} prime(k)*(exp(x) - 1)^k/k!.
%C Stirling transform of primes.
%H Alois P. Heinz, <a href="/A307771/b307771.txt">Table of n, a(n) for n = 1..574</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F G.f.: Sum_{k>=1} prime(k)*x^k / Product_{j=1..k} (1 - j*x).
%F a(n) = Sum_{k=1..n} Stirling2(n,k)*prime(k).
%p a:= n-> add(ithprime(k)*Stirling2(n, k), k=1..n):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Apr 27 2019
%p # second Maple program:
%p b:= proc(n, m) option remember;
%p `if`(n=0, ithprime(m), m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n-1, 1):
%p seq(a(n), n=1..24); # _Alois P. Heinz_, Aug 03 2021
%t nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
%t nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[Sum[StirlingS2[n, k] Prime[k], {k, 1, n}], {n, 1, 24}]
%Y Cf. A000040, A085507, A307772, A307773.
%K nonn
%O 1,1
%A _Ilya Gutkovskiy_, Apr 27 2019
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