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Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).
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%I #5 Feb 16 2025 08:33:37

%S 1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0,

%T 1,2,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0,0,1,2,1,0,1,2,1,0,0,0,1,2,1,

%U 0,1,2,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,1,1,1,2,1,0,1,2,1,0,1,1,0,1,1,0,1,2,1,0,1,1,0,1,1,0,1,2,1,0,1,2,2

%N Expansion of Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

%C Number of partitions of n into distinct tetrahedral numbers (A000292).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} (1 + x^(k*(k+1)*(k+2)/6)).

%e a(35) = 2 because we have [35] and [20, 10, 4, 1].

%t nmax=120; CoefficientList[Series[Product[1 + x^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000292, A007294, A024940, A068980.

%K nonn,changed

%O 0,36

%A _Ilya Gutkovskiy_, Dec 09 2016