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

1, 1, 4, 10, 29, 72, 200, 510, 1364, 3546, 9348, 24400, 64090, 167562, 439200, 1149360, 3010349, 7879832, 20633304, 54014950, 141422328, 370239300, 969323000, 2537696160, 6643839400, 17393731933, 45537549048, 119218684970, 312119004990, 817137724392, 2139295489200, 5600747143950

Offset: 0

Author

Ilya Gutkovskiy, Nov 06 2019

Keywords

• nonn

Comments

Euler transform of A032198.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2392
• Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.

Crossrefs

Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163, A386706, A387017.

Convolution inverse of A329157.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(a(j)*b(n-j$2), j=0..n-1))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 25 2025

Mathematica

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).

G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).

G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.

a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019

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