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A321139
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a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).
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4
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1, 1, 1, 3, 7, 17, 52, 144, 480, 1732, 5902, 21078, 78434, 289107, 1079949, 4094643, 15574377, 59667023, 230318968, 892694240, 3477119540, 13606993083, 53438614380, 210622413188, 832922044686, 3303392730698, 13137474884294, 52381331536536, 209340904575968
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OFFSET
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0,4
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COMMENTS
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Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.
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LINKS
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FORMULA
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a(n) = [x^(n^2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.
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EXAMPLE
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1*0^2 + 2*1^2 + 3*1^2 + 4*0^2 + 5*2^2 = 25.
1*0^2 + 2*2^2 + 3*2^2 + 4*0^2 + 5*1^2 = 25.
1*0^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*1^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*2^2 = 25.
1*1^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*1^2 = 25.
1*1^2 + 2*2^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*1^2 + 2*2^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*2^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*1^2 = 25.
1*2^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*1^2 = 25.
1*2^2 + 2*1^2 + 3*1^2 + 4*2^2 + 5*0^2 = 25.
1*2^2 + 2*3^2 + 3*1^2 + 4*0^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*0^2 + 4*2^2 + 5*0^2 = 25.
1*3^2 + 2*0^2 + 3*2^2 + 4*1^2 + 5*0^2 = 25.
1*3^2 + 2*2^2 + 3*1^2 + 4*0^2 + 5*1^2 = 25.
1*4^2 + 2*0^2 + 3*0^2 + 4*1^2 + 5*1^2 = 25.
1*4^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*0^2 = 25.
1*5^2 + 2*0^2 + 3*0^2 + 4*0^2 + 5*0^2 = 25.
So a(5) = 17.
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MAPLE
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b:= proc(n, i) option remember; local j; if n=0 then 1
elif i<1 then 0 else b(n, i-1); for j while
i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
end:
a:= n-> b(n^2, n):
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MATHEMATICA
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nmax = 25; Table[SeriesCoefficient[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)
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PROG
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(PARI) {a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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