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A299032
Number of ordered ways of writing n-th triangular number as a sum of n squares of positive integers.
3
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
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}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 01 2018
STATUS
approved