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

%I A380614 #7 Feb 16 2025 08:34:07
%S A380614 2,5,20,155,1860,24970,444060,8583935,202071920,5992773714,
%T A380614 186947632200,7001535728810,288868991951760,12455290280871150,
%U A380614 587972068547997856,31327583556949986095,1856116108295418943020,113366872636395467452840,7619343577986975410930880,541957669076266404650853414
%N A380614 Product_{n>=1} (1 + x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.
%C A380614 Inverse Weigh transform of primorial numbers.
%H A380614 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%p A380614 p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
%p A380614 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A380614       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A380614     end:
%p A380614 a:= proc(n) option remember; p(n)-b(n, n-1) end:
%p A380614 seq(a(n), n=1..20);  # _Alois P. Heinz_, Jan 28 2025
%t A380614 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]
%Y A380614 Cf. A002110, A168246, A305871, A380498, A380613.
%K A380614 nonn
%O A380614 1,1
%A A380614 _Ilya Gutkovskiy_, Jan 28 2025