A353079 Exponential transform of odd primes.

1, 3, 14, 79, 521, 3876, 31935, 287225, 2791122, 29066589, 322292257, 3784650052, 46857941291, 609360372095, 8296220760974, 117914344818807, 1745211622467633, 26838798853062516, 428009369349905497, 7065576909286562195, 120545067517808693300, 2122393931891338237325, 38512344746420591905771

Offset: 0

Author

Ilya Gutkovskiy, Apr 22 2022

Keywords

• nonn

Crossrefs

Cf. A007446, A065091, A300632.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j+1)*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 27 2022

Mathematica

nmax = 22; CoefficientList[Series[Exp[Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * prime(k+1) * a(n-k).