A300632 Expansion of e.g.f. exp(x + Sum_{k>=2} prime(k-1)*x^k/k!).

1, 1, 3, 10, 42, 203, 1119, 6839, 45895, 334142, 2619052, 21946647, 195537777, 1843619725, 18321431155, 191242913022, 2090436115146, 23864653888881, 283865214366771, 3510656353388517, 45056394441558593, 599057016471131604, 8238406603745152620, 117020080948487107289

Offset: 0

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

• nonn

Comments

Exponential transform of A008578.

Examples

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 42*x^4/4! + 203*x^5/5! + 1119*x^6/6! + 6839*x^7/7! + ..
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..534
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000040, A007446, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016.

Programs

Maple

a:= proc(n) option remember; (p-> `if`(n=0, 1, add(a(n-j)*p(j)*
      binomial(n-1, j-1), j=1..n)))(t-> `if`(t=1, 1, ithprime(t-1)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018

Mathematica

nmax = 23; CoefficientList[Series[Exp[x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} A008578(k)*x^k/k!).