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A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).

%I A335633 #11 Feb 16 2025 08:34:00
%S A335633 1,1,3,6,5,95,336,2597,26832,197577,1847800,14621101,129754956,
%T A335633 1146534701,12342194879,161225146370,2464561564936,39642413790129,
%U A335633 620059254486798,9430493858327959,136438759335452360,1881721996407396801,24999081626667425376,321601467988647184779
%N A335633 Number of ordered ways of writing the n-th n-gonal number as a sum of n n-gonal numbers (with 0's allowed).
%H A335633 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%H A335633 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A335633 a(n) = [x^p(n,n)] (Sum_{k=0..n} x^p(n,k))^n, where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.
%e A335633 a(3) = 6 because the third triangular number is 6 and we have [6, 0, 0], [0, 6, 0], [0, 0, 6], [3, 3, 0], [3, 0, 3] and [0, 3, 3].
%t A335633 Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 0, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 0, 23}]
%o A335633 (PARI)
%o A335633 p(n,k) = {k * (k * (n - 2) - n + 4) / 2}
%o A335633 a(n) = {my(m=p(n,n)); polcoef((sum(k=0, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ _Andrew Howroyd_, Oct 03 2020
%Y A335633 Cf. A060354, A088218, A196010, A298329, A335634, A337762, A337763, A337764.
%K A335633 nonn
%O A335633 0,3
%A A335633 _Ilya Gutkovskiy_, Oct 03 2020