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A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

%I A329156 #30 Aug 20 2025 10:57:23
%S A329156 1,1,4,10,29,72,200,510,1364,3546,9348,24400,64090,167562,439200,
%T A329156 1149360,3010349,7879832,20633304,54014950,141422328,370239300,
%U A329156 969323000,2537696160,6643839400,17393731933,45537549048,119218684970,312119004990,817137724392,2139295489200,5600747143950
%N A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).
%C A329156 Euler transform of A032198.
%H A329156 Alois P. Heinz, <a href="/A329156/b329156.txt">Table of n, a(n) for n = 0..2392</a>
%H A329156 Christian Kassel and Christophe Reutenauer, <a href="https://arxiv.org/abs/2507.15780">Pairs of intertwined integer sequences</a>, arXiv:2507.15780 [math.NT], 2025. See p. 13.
%F A329156 G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).
%F A329156 G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
%F A329156 G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).
%F A329156 G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.
%F A329156 a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 07 2019
%F A329156 a(2^k) = A002878(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 on page 11 of Kassel-Reutenauer paper. - _Michael De Vlieger_, Jul 28 2025
%p A329156 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
%p A329156       add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
%p A329156     end:
%p A329156 a:= proc(n) option remember; `if`(n=0, 1,
%p A329156       -add(a(j)*b(n-j$2), j=0..n-1))
%p A329156     end:
%p A329156 seq(a(n), n=0..31);  # _Alois P. Heinz_, Jul 25 2025
%t A329156 nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
%t A329156 nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A329156 Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163, A386706, A387017.
%Y A329156 Convolution inverse of A329157.
%K A329156 nonn
%O A329156 0,3
%A A329156 _Ilya Gutkovskiy_, Nov 06 2019