A280351 Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).
1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3433, 6436, 11441, 19449, 31825, 50389, 77521, 116281, 170545, 245158, 346105, 480701, 657802, 888058, 1184419, 1564435, 2063206, 2799487, 4272049, 8544097, 23535821, 77331981, 262534537, 865287625, 2720095405
Offset: 0
Examples
a(9) = 9 because we have: [1] [9] [2] [2, 1, 1, 1, 1, 1, 1, 1] [3] [1, 2, 1, 1, 1, 1, 1, 1] [4] [1, 1, 2, 1, 1, 1, 1, 1] [5] [1, 1, 1, 2, 1, 1, 1, 1] [6] [1, 1, 1, 1, 2, 1, 1, 1] [7] [1, 1, 1, 1, 1, 2, 1, 1] [8] [1, 1, 1, 1, 1, 1, 2, 1] [9] [1, 1, 1, 1, 1, 1, 1, 2]
Programs
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Maple
a := n -> ifelse(n = 0, 1, add(binomial(n - 1, k^3 - 1), k = 1..floor(n^(1/3)))): seq(a(n), n = 0..39); # Peter Luschny, Dec 23 2022
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Mathematica
nmax = 39; CoefficientList[Series[Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]
Formula
a(0) = 1; a(n) = Sum_{k=1..floor(n^(1/3))} binomial(n-1, k^3-1) for n > 0. - Jerzy R Borysowicz, Dec 22 2022
Extensions
a(0)=1 prepended by Alois P. Heinz, Dec 17 2022
Comments