%I #34 Nov 29 2023 07:42:52
%S 1,1,4,4,14,14,40,40,106,106,254,254,582,582,1256,1256,2620,2620,5256,
%T 5256,10266,10266,19482,19482,36204,36204,65792,65792,117496,117496,
%U 206120,206120,356320,356320,606912,606912,1020848,1020848,1695676,1695676,2786010
%N Expansion of g.f. Product_{k>=2} 1/(1-x^phi(k)).
%H Vaclav Kotesovec, <a href="/A347428/b347428.txt">Table of n, a(n) for n = 0..10000</a>
%H David P. Roberts and Fernando Rodriguez Villegas, <a href="https://arxiv.org/abs/2109.00027">Hypergeometric Motives</a>, arXiv:2109.00027 [math.AG], 2021. See (5.2) p. 6.
%F From _Vaclav Kotesovec_, Sep 02 2021: (Start)
%F For n>0, a(n) = A120963(n) - A120963(n-1).
%F log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. (End)
%p with(numtheory):
%p b:= proc(n) option remember; nops(invphi(n)) end:
%p g:= proc(n) option remember; `if`(n=0, 1, add(
%p g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
%p end:
%p a:= n-> g(n)-g(n-1):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 23 2023
%t nt = 100; (* number of terms *)
%t f[kmax_] := f[kmax] = CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 2, kmax}] + O[x]^nt, x]; f[kmax = nt]; f[kmax += nt];
%t While[f[kmax] != f[kmax - nt], kmax += nt];
%t f[kmax] (* _Jean-François Alcover_, Nov 29 2023 *)
%Y Cf. A000010 (phi), A014197, A051894, A120963 (similar g.f.).
%K nonn
%O 0,3
%A _Michel Marcus_, Sep 02 2021
%E Terms a(16) and beyond corrected by _Vaclav Kotesovec_, Jun 23 2023, following a suggestion from _Georg Fischer_