A335634 - OEIS

%I #11 Feb 16 2025 08:34:00

%S 1,1,1,0,1,30,180,700,3780,11844,50610,325820,5803380,126594910,

%T 2114901789,28282722650,323420067880,3190581939996,29336527986960,

%U 245438739897312,1967485926594030,16000631392009320,184418174847183508,4054670001158799616,111835386569787369559

%N Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.

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

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%F a(n) = [x^p(n,n)] (Sum_{k=1..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 a(4) = 1 because the fourth square is 16 and we have [4, 4, 4, 4].

%t Join[{1}, Table[SeriesCoefficient[Sum[x^(k (k (n - 2) - n + 4)/2), {k, 1, n}]^n, {x, 0, n (n^2 - 3 n + 4)/2}], {n, 1, 24}]]

%o (PARI)

%o p(n,k) = {k * (k * (n - 2) - n + 4) / 2}

%o a(n) = {my(m=p(n,n)); polcoef((sum(k=1, n, x^p(n,k)) + O(x*x^m))^n, m)} \\ _Andrew Howroyd_, Oct 03 2020

%Y Cf. A060354, A298330, A298858, A335633, A337762, A337763, A337764.

%K nonn,changed

%O 0,6

%A _Ilya Gutkovskiy_, Oct 03 2020