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A336091 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n n-gonal pyramidal numbers (with 0's allowed).

%I A336091 #9 Feb 16 2025 08:34:00
%S A336091 1,1,2,3,10,5,246,1519,19678,74601,690490,21026621,301528272,
%T A336091 4397123315,71221546592,1001245733295,19276579678736,368677642975493,
%U A336091 6820451221691646,136000924000323691,3069656935024721420,69646109074231173897,1641880679174919030100
%N A336091 Number of ordered ways of writing the n-th n-gonal pyramidal number as a sum of n n-gonal pyramidal numbers (with 0's allowed).
%H A336091 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>
%H A336091 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A336091 a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k + 1) * (k * (n - 2) - n + 5) / 6 is the k-th n-gonal pyramidal number.
%e A336091 a(3) = 3 because the third tetrahedral (or triangular pyramidal) number is 10 and we have [10, 0, 0], [0, 10, 0] and [0, 0, 10].
%t A336091 Table[SeriesCoefficient[Sum[x^(k (k + 1) (k (n - 2) - n + 5)/6), {k, 0, n}]^n, {x, 0, n (n + 1) (n^2 - 3 n + 5)/6}], {n, 0, 22}]
%Y A336091 Cf. A006484, A196010, A335633, A336303, A337797, A337798, A337799.
%K A336091 nonn
%O A336091 0,3
%A A336091 _Ilya Gutkovskiy_, Oct 04 2020