A307771 - OEIS

%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