A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.

1, 1, 0, 3, 6, 0, 12, 106, 420, 2718, 18240, 120879, 694320, 5430438, 40668264, 300401818, 2369504386, 19928714475, 174151735920, 1543284732218, 14224347438876, 135649243229688, 1331658133954940, 13369350846412794, 138122850643702056, 1462610254141337590

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Index entries for sequences related to sums of squares

Index to sequences related to polygonal numbers

Formula

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.

Example

a(4) = 6 because fourth triangular number is 10 and we have [4, 4, 1, 1], [4, 1, 4, 1], [4, 1, 1, 4], [1, 4, 4, 1], [1, 4, 1, 4] and [1, 1, 4, 4].

Maple

b:= proc(n, t) option remember; local i; if n=0 then
      `if`(t=0, 1, 0) elif t<1 then 0 else 0;
      for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
    end:
a:= n-> b(n*(n+1)/2, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018

Mathematica

Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]

Crossrefs

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031.

Sequence in context: A287845 A070297 A284896 * A094674 A125287 A264813

Adjacent sequences: A299029 A299030 A299031 * A299033 A299034 A299035

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 01 2018

Status

approved