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A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

%I A337762 #22 Feb 16 2025 08:34:00
%S A337762 1,1,2,4,8,21,56,144,370,926,2275,5482,12966,30124,68838,154934,
%T A337762 343756,752689,1627701,3479226,7355608,15390682,31889732,65465473,
%U A337762 133212912,268811363,538119723,1069051243,2108416588,4129355331,8033439333
%N A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.
%H A337762 David A. Corneth, <a href="/A337762/b337762.txt">Table of n, a(n) for n = 0..278</a> (first 51 terms from Vaclav Kotesovec)
%H A337762 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%H A337762 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A337762 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A337762 a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.
%e A337762 a(3) = 4 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].
%t A337762 nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^(k*((k*(n - 2) - n + 4)/2))), {k, 1, n}], {x, 0, n*(4 - 3*n + n^2)/2}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Sep 19 2020 *)
%Y A337762 Cf. A000041, A037444, A060354, A072964, A337763, A337764.
%K A337762 nonn
%O A337762 0,3
%A A337762 _Ilya Gutkovskiy_, Sep 19 2020