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Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).
3

%I #41 Jan 22 2024 15:22:41

%S 1,1,1,1,1,1,1,1,2,9,37,121,331,793,1717,3433,6436,11441,19449,31825,

%T 50389,77521,116281,170545,245158,346105,480701,657802,888058,1184419,

%U 1564435,2063206,2799487,4272049,8544097,23535821,77331981,262534537,865287625,2720095405

%N Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).

%C Number of compositions of n into a cube number of parts.

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F 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

%e a(9) = 9 because we have:

%e [1] [9]

%e [2] [2, 1, 1, 1, 1, 1, 1, 1]

%e [3] [1, 2, 1, 1, 1, 1, 1, 1]

%e [4] [1, 1, 2, 1, 1, 1, 1, 1]

%e [5] [1, 1, 1, 2, 1, 1, 1, 1]

%e [6] [1, 1, 1, 1, 2, 1, 1, 1]

%e [7] [1, 1, 1, 1, 1, 2, 1, 1]

%e [8] [1, 1, 1, 1, 1, 1, 2, 1]

%e [9] [1, 1, 1, 1, 1, 1, 1, 2]

%p a := n -> ifelse(n = 0, 1, add(binomial(n - 1, k^3 - 1), k = 1..floor(n^(1/3)))):

%p seq(a(n), n = 0..39); # _Peter Luschny_, Dec 23 2022

%t nmax = 39; CoefficientList[Series[Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A000578, A052467, A103198, A280352.

%K nonn,easy

%O 0,9

%A _Ilya Gutkovskiy_, Jan 01 2017

%E a(0)=1 prepended by _Alois P. Heinz_, Dec 17 2022