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A321139 a(n) = [x^(n^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 101 terms from Seiichi Manyama)

FORMULA

a(n) = [x^(n^2)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.

EXAMPLE

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.

MAPLE

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):

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 28 2018

MATHEMATICA

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 *)

PROG

(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)}

CROSSREFS

Cf. A000122, A010052, A206226, A300446, A320932.

Sequence in context: A324789 A014144 A247183 * A096358 A260349 A146147

Adjacent sequences:  A321136 A321137 A321138 * A321140 A321141 A321142

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 28 2018

STATUS

approved

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Last modified April 17 11:50 EDT 2021. Contains 343064 sequences. (Running on oeis4.)