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%I #26 Oct 29 2018 12:38:40
%S 1,2,4,12,24,80,292,966,3876,15554,61608,254612,1065676,4471672,
%T 19074968,82043172,354365492,1543432514,6760146292,29732837780,
%U 131440491584,583419967664,2598585783488,11615321544700,52079369904384,234157152231726,1055628140278948,4770576024205060
%N a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.
%C Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n*(n+1)/2.
%H Vaclav Kotesovec, <a href="/A320931/b320931.txt">Table of n, a(n) for n = 0..300</a> (first 101 terms from Seiichi Manyama)
%F a(n) ~ c * d^n / n^(7/4), where d = 4.818071572655... and c = 0.5869031198... - _Vaclav Kotesovec_, Oct 29 2018
%e Solutions (a_1, a_2, ... , a_4) to the equation a_1^2 + 2*a_2^2 + ... + 4*a_4^2 = 10.
%e -------------------------------------------------------------------------------------
%e ( 1, 1, 1, 1), ( 1, 1, 1, -1),
%e ( 1, 1, -1, 1), ( 1, 1, -1, -1),
%e ( 1, -1, 1, 1), ( 1, -1, 1, -1),
%e ( 1, -1, -1, 1), ( 1, -1, -1, -1),
%e (-1, 1, 1, 1), (-1, 1, 1, -1),
%e (-1, 1, -1, 1), (-1, 1, -1, -1),
%e (-1, -1, 1, 1), (-1, -1, 1, -1),
%e (-1, -1, -1, 1), (-1, -1, -1, -1),
%e ( 2, 1, 0, 1), ( 2, 1, 0, -1),
%e ( 2, -1, 0, 1), ( 2, -1, 0, -1),
%e (-2, 1, 0, 1), (-2, 1, 0, -1),
%e (-2, -1, 0, 1), (-2, -1, 0, -1).
%t nmax = 25; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n*(n+1)/2}], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 29 2018 *)
%Y Cf. A000122, A000217, A173519, A320067, A320932.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Oct 28 2018
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