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

%I A299032 #7 Feb 05 2018 12:49:58
%S A299032 1,1,0,3,6,0,12,106,420,2718,18240,120879,694320,5430438,40668264,
%T A299032 300401818,2369504386,19928714475,174151735920,1543284732218,
%U A299032 14224347438876,135649243229688,1331658133954940,13369350846412794,138122850643702056,1462610254141337590
%N A299032 Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.
%H A299032 Alois P. Heinz, <a href="/A299032/b299032.txt">Table of n, a(n) for n = 0..250</a>
%H A299032 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%H A299032 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A299032 a(n) = [x^(n*(n+1)/2)] (Sum_{k>=1} x^(k^2))^n.
%e A299032 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].
%p A299032 b:= proc(n, t) option remember; local i; if n=0 then
%p A299032       `if`(t=0, 1, 0) elif t<1 then 0 else 0;
%p A299032       for i while i^2<=n do %+b(n-i^2, t-1) od; % fi
%p A299032     end:
%p A299032 a:= n-> b(n*(n+1)/2, n):
%p A299032 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 05 2018
%t A299032 Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 25}]
%Y A299032 Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298330, A298858, A298939, A299031.
%K A299032 nonn
%O A299032 0,4
%A A299032 _Ilya Gutkovskiy_, Feb 01 2018