A319758 Expansion of Product_{k>=1} 1/(1 - Sum_{j=1..k} x^(j*k)).

1, 1, 2, 3, 6, 8, 15, 20, 34, 48, 76, 103, 165, 222, 335, 461, 683, 919, 1352, 1813, 2611, 3519, 4985, 6651, 9408, 12501, 17401, 23165, 32009, 42312, 58241, 76748, 104725, 138017, 187155, 245521, 332135, 434536, 584023, 763799, 1022507, 1332549, 1779534, 2314437, 3077540, 3999825

Offset: 0

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A052335, A115584.

Programs

Maple

a:=series(mul(1/(1-add(x^(j*k),j=1..k)),k=1..100),x=0,46): seq(coeff(a,x,n),n=0..45); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 45; CoefficientList[Series[Product[1/(1 - Sum[x^(j k), {j, 1, k}]), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

From Vaclav Kotesovec, Sep 27 2018: (Start)

a(n) ~ c * phi^(n/2), where

c = 188.4773924093125890061786423020365148584841831715... if n is even

c = 187.5693962190327254176348797865060646998844995050... if n is odd

phi = A001622 = (1+sqrt(5))/2 is the golden ratio. (End)