A351681 -id:A351681 - cp's OEIS frontend

A353406 Stirling transform of odd primes.

3, 8, 25, 91, 376, 1715, 8471, 44838, 252903, 1514213, 9590874, 64056173, 449804453, 3312346950, 25521479277, 205300781275, 1720450321356, 14986361037495, 135393159641569, 1266006310597506, 12228936468908781, 121823473948915769, 1249794986354577736

Offset: 1

Ilya Gutkovskiy, May 07 2022

nonn

Cf. A065091, A307771, A351681.

b:= proc(n, m) option remember;
     `if`(n=0, ithprime(m+1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n-1, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, May 13 2022

nmax = 23; CoefficientList[Series[Sum[Prime[k + 1] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] Prime[k + 1], {k, 1, n}], {n, 1, 23}]

E.g.f.: Sum_{k>=1} prime(k+1) * (exp(x) - 1)^k / k!.

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

A354003 Inverse Stirling transform of A008578 (1 together with the primes).

Original entry on oeis.org

1, 1, -1, 3, -14, 84, -604, 5020, -47144, 492408, -5653004, 70681706, -955450018, 13878511166, -215521103888, 3562431678650, -62439880637498, 1156609714838858, -22575425757129216, 463085375385002432, -9959296414838153618, 224079866356625633070, -5264190202707104532482

Offset: 1

Ilya Gutkovskiy, May 13 2022

sign

Cf. A008578, A307772, A307773, A351681, A354002.

nmax = 23; CoefficientList[Series[Log[1 + x] + Sum[Prime[k - 1] Log[1 + x]^k/k!, {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] If[k == 1, 1, Prime[k - 1]], {k, 1, n}], {n, 1, 23}]

E.g.f.: log(1 + x) + Sum_{k>=2} prime(k-1) * log(1 + x)^k / k!.

a(n) = Sum_{k=1..n} Stirling1(n,k) * A008578(k).

cp's OEIS Frontend

A353406 Stirling transform of odd primes.

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