cp's OEIS Frontend

A307771 Expansion of e.g.f. Sum_{k>=1} prime(k)*(exp(x) - 1)^k/k!.

%I A307771 #20 Mar 30 2025 04:28:56
%S A307771 2,5,16,60,253,1178,5976,32623,189702,1166720,7554877,51351254,
%T A307771 365560784,2720255911,21121563036,170839106566,1437200307921,
%U A307771 12556366592382,113755900474652,1067028469382353,10346222830388738,103538470949470066,1067747451140472577,11330777204488565252
%N A307771 Expansion of e.g.f. Sum_{k>=1} prime(k)*(exp(x) - 1)^k/k!.
%C A307771 Stirling transform of primes.
%H A307771 Alois P. Heinz, <a href="/A307771/b307771.txt">Table of n, a(n) for n = 1..574</a>
%H A307771 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A307771 G.f.: Sum_{k>=1} prime(k)*x^k / Product_{j=1..k} (1 - j*x).
%F A307771 a(n) = Sum_{k=1..n} Stirling2(n,k)*prime(k).
%p A307771 a:= n-> add(ithprime(k)*Stirling2(n, k), k=1..n):
%p A307771 seq(a(n), n=1..30);  # _Alois P. Heinz_, Apr 27 2019
%p A307771 # second Maple program:
%p A307771 b:= proc(n, m) option remember;
%p A307771      `if`(n=0, ithprime(m), m*b(n-1, m)+b(n-1, m+1))
%p A307771     end:
%p A307771 a:= n-> b(n-1, 1):
%p A307771 seq(a(n), n=1..24);  # _Alois P. Heinz_, Aug 03 2021
%t A307771 nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
%t A307771 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 A307771 Table[Sum[StirlingS2[n, k] Prime[k], {k, 1, n}], {n, 1, 24}]
%o A307771 (PARI) upto(n) = my(v1, v2); v1 = vector(n, i, prime(i)); v2 = vector(n, i, 0); v2[1] = v1[1]; for(i=1, n-1, v1 = vector(#v1-1, j, j*v1[j] + v1[j+1]); v2[i+1] = v1[1]); v2 \\ _Mikhail Kurkov_, Mar 30 2025, after _Alois P. Heinz_
%Y A307771 Cf. A000040, A085507, A307772, A307773.
%K A307771 nonn
%O A307771 1,1
%A A307771 _Ilya Gutkovskiy_, Apr 27 2019