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

2, 5, 16, 60, 253, 1178, 5976, 32623, 189702, 1166720, 7554877, 51351254, 365560784, 2720255911, 21121563036, 170839106566, 1437200307921, 12556366592382, 113755900474652, 1067028469382353, 10346222830388738, 103538470949470066, 1067747451140472577, 11330777204488565252

Offset: 1

Author

Ilya Gutkovskiy, Apr 27 2019

Keywords

• nonn

Comments

Stirling transform of primes.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..574
• Eric Weisstein's World of Mathematics, Stirling Transform

Crossrefs

Cf. A000040, A085507, A307772, A307773.

Programs

Maple

a:= n-> add(ithprime(k)*Stirling2(n, k), k=1..n):
seq(a(n), n=1..30);  # Alois P. Heinz, Apr 27 2019
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, ithprime(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n-1, 1):
seq(a(n), n=1..24);  # Alois P. Heinz, Aug 03 2021

Mathematica

nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[Prime[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] Prime[k], {k, 1, n}], {n, 1, 24}]

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

Formula

G.f.: Sum_{k>=1} prime(k)*x^k / Product_{j=1..k} (1 - j*x).

a(n) = Sum_{k=1..n} Stirling2(n,k)*prime(k).