A335634 Number of ordered ways of writing the n-th n-gonal number as a sum of n nonzero n-gonal numbers.
1, 1, 1, 0, 1, 30, 180, 700, 3780, 11844, 50610, 325820, 5803380, 126594910, 2114901789, 28282722650, 323420067880, 3190581939996, 29336527986960, 245438739897312, 1967485926594030, 16000631392009320, 184418174847183508, 4054670001158799616, 111835386569787369559
Offset: 0
Keywords
Examples
a(4) = 1 because the fourth square is 16 and we have [4, 4, 4, 4].
Links
- Eric Weisstein's World of Mathematics, Polygonal Number
- Index to sequences related to polygonal numbers
Programs
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Mathematica
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}]] -
PARI
p(n,k) = {k * (k * (n - 2) - n + 4) / 2} 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
Formula
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.