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A380613 Expansion of Product_{k>=1} (1 + x^k)^prime(k)#.

%I A380613 #8 Feb 16 2025 08:34:07
%S A380613 1,2,7,42,291,2970,36950,597100,11070875,248103940,7018494836,
%T A380613 215718595582,7881561212732,320881902092122,13754717161317416,
%U A380613 643588827524430916,33926485821837232397,1992916854095359256932,121393059052727838936847,8107963745977267426512386,574571379331620422000295082
%N A380613 Expansion of Product_{k>=1} (1 + x^k)^prime(k)#.
%C A380613 Weigh transform of primorial numbers.
%H A380613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%p A380613 p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
%p A380613 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A380613       add(binomial(p(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A380613     end:
%p A380613 a:= n-> b(n$2):
%p A380613 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 28 2025
%t A380613 nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Product[Prime[j], {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]
%t A380613 primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(j/d + 1) d primorial[d], {d, Divisors[j]}] a[n - j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}]
%Y A380613 Cf. A002110, A061152, A261052, A380497, A380614.
%K A380613 nonn
%O A380613 0,2
%A A380613 _Ilya Gutkovskiy_, Jan 28 2025