A300661 Expansion of e.g.f. exp(-Sum_{k>=1} prime(k)*x^k/k!).

1, -2, 1, 5, 4, -53, -177, 282, 5759, 20355, -83420, -1420133, -6245485, 29035652, 648899541, 4034393367, -10488623858, -464971765297, -4310935438663, -3489419105786, 446500913437911, 6423072226704027, 30987397708208720, -462727554963927783, -11862200720684515159

Offset: 0

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

• sign

Examples

			E.g.f.: A(x) = 1 - 2*x/1! + x^2/2! + 5*x^3/3! + 4*x^4/4! - 53*x^5/5! - 177*x^6/6! + 282*x^7/7! + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..542
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000040, A007446, A007447, A030017, A030018, A300632.

Programs

Maple

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

Mathematica

nmax = 24; CoefficientList[Series[Exp[-Sum[Prime[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[-Prime[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: exp(-Sum_{k>=1} A000040(k)*x^k/k!).