A380614 Product_{n>=1} (1 + x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.

2, 5, 20, 155, 1860, 24970, 444060, 8583935, 202071920, 5992773714, 186947632200, 7001535728810, 288868991951760, 12455290280871150, 587972068547997856, 31327583556949986095, 1856116108295418943020, 113366872636395467452840, 7619343577986975410930880, 541957669076266404650853414

Offset

1,1

Comments

Inverse Weigh transform of primorial numbers.

Links

Table of n, a(n) for n=1..20.

Eric Weisstein's World of Mathematics, Primorial.

Maple

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; p(n)-b(n, n-1) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 28 2025

Mathematica

primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = primorial[n] - b[n, n - 1]; Array[a, 20]

Crossrefs

Cf. A002110, A168246, A305871, A380498, A380613.

Sequence in context: A136650 A229662 A111885 * A159320 A184730 A181076

Adjacent sequences: A380611 A380612 A380613 * A380615 A380616 A380617

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 28 2025

Status

approved