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A307771
Expansion of e.g.f. Sum_{k>=1} prime(k)*(exp(x) - 1)^k/k!.
6
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
1,1
COMMENTS
Stirling transform of primes.
LINKS
Eric Weisstein's World of Mathematics, Stirling Transform
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).
MAPLE
a:= n-> add(ithprime(k)*Stirling2(n, k), k=1..n):
seq(a(n), n=1..30); # Alois P. Heinz, Apr 27 2019
# Alternative:
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}]
PROG
(PARI) list(n) = my(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 27 2019
STATUS
approved