A030017 a(1) = 1, a(n+1) = Sum_{k = 1..n} p(k)*a(n+1-k), where p(k) is the k-th prime.

1, 2, 7, 25, 88, 311, 1095, 3858, 13591, 47881, 168688, 594289, 2093693, 7376120, 25986209, 91549913, 322532092, 1136286727, 4003159847, 14103208628, 49685873471, 175044281583, 616684348614, 2172590743211, 7654078700221, 26965465508072, 94999850216565

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn
• nice

Comments

Apply "INVERT" transform to primes.

Examples

			a(5) = 25*2 +7*3 +2*5 + 1*7 = 88.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
• N. J. A. Sloane, Transforms

Crossrefs

Row sums of A340991(n-1).

Cf. A000040.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add(a(n-i)*ithprime(i), i=1..n-1))
    end:
seq(a(n), n=1..29);  # Alois P. Heinz, Feb 10 2021

Mathematica

CoefficientList[ Series[ 1/(1 - Sum[ Prime[ n ]*x^n, {n, 1, 25} ] ), {x, 0, 25} ], x ]
(* Second program: *)
a[1] = 1; a[m_] := a[m] = Sum[Prime@ k  a[m - k], {k, m - 1}]; Table[a@ n, {n, 25}] (* Michael De Vlieger, Dec 13 2016 *)

Formula

INVERT: a's from b's in 1+Sum a_i x^i = 1/(1-Sum b_i x^i).

G.f: (1-b(x)/(b(x)-1))*x, where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 13 2016