A321208 a(n) = [x^(n*(n+1)*(2*n+1)/6)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

1, 1, 0, 2, 7, 31, 167, 1046, 7949, 60487, 490753, 4232323, 39877499, 401064825, 4191449438, 45993709856, 526379057073, 6284584514360, 77594053714675, 990497759689341, 13053609492660678, 177385290308586391, 2480368806876623852, 35617209442716039028, 524705024124493308382

Offset: 0

Author

Seiichi Manyama, Oct 30 2018

Keywords

• nonn

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*(n+1)*(2*n+1)/6.

Examples

			1*0^2 + 2*1^2 + 3*2^2 + 4*3^2 + 5*1^2 = 55.
1*0^2 + 2*1^2 + 3*4^2 + 4*0^2 + 5*1^2 = 55.
1*0^2 + 2*2^2 + 3*3^2 + 4*0^2 + 5*2^2 = 55.
1*0^2 + 2*4^2 + 3*1^2 + 4*0^2 + 5*2^2 = 55.
1*0^2 + 2*5^2 + 3*0^2 + 4*0^2 + 5*1^2 = 55.
1*1^2 + 2*1^2 + 3*1^2 + 4*1^2 + 5*3^2 = 55.
1*1^2 + 2*1^2 + 3*4^2 + 4*1^2 + 5*0^2 = 55.
1*1^2 + 2*3^2 + 3*0^2 + 4*2^2 + 5*2^2 = 55.
1*1^2 + 2*3^2 + 3*0^2 + 4*3^2 + 5*0^2 = 55.
1*1^2 + 2*3^2 + 3*2^2 + 4*1^2 + 5*2^2 = 55.
1*1^2 + 2*3^2 + 3*3^2 + 4*1^2 + 5*1^2 = 55.
1*1^2 + 2*5^2 + 3*0^2 + 4*1^2 + 5*0^2 = 55.
1*2^2 + 2*0^2 + 3*3^2 + 4*1^2 + 5*2^2 = 55.
1*2^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*3^2 = 55.
1*2^2 + 2*2^2 + 3*3^2 + 4*2^2 + 5*0^2 = 55.
1*2^2 + 2*3^2 + 3*2^2 + 4*2^2 + 5*1^2 = 55.
1*2^2 + 2*4^2 + 3*1^2 + 4*2^2 + 5*0^2 = 55.
1*3^2 + 2*1^2 + 3*1^2 + 4*3^2 + 5*1^2 = 55.
1*3^2 + 2*3^2 + 3*2^2 + 4*2^2 + 5*0^2 = 55.
1*4^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*2^2 = 55.
1*4^2 + 2*0^2 + 3*1^2 + 4*3^2 + 5*0^2 = 55.
1*4^2 + 2*2^2 + 3*3^2 + 4*1^2 + 5*0^2 = 55.
1*4^2 + 2*3^2 + 3*0^2 + 4*2^2 + 5*1^2 = 55.
1*4^2 + 2*3^2 + 3*2^2 + 4*1^2 + 5*1^2 = 55.
1*4^2 + 2*4^2 + 3*1^2 + 4*1^2 + 5*0^2 = 55.
1*5^2 + 2*1^2 + 3*2^2 + 4*2^2 + 5*0^2 = 55.
1*5^2 + 2*3^2 + 3*1^2 + 4*1^2 + 5*1^2 = 55.
1*5^2 + 2*3^2 + 3*2^2 + 4*0^2 + 5*0^2 = 55.
1*6^2 + 2*0^2 + 3*1^2 + 4*2^2 + 5*0^2 = 55.
1*6^2 + 2*1^2 + 3*2^2 + 4*0^2 + 5*1^2 = 55.
1*7^2 + 2*1^2 + 3*0^2 + 4*1^2 + 5*0^2 = 55.
So a(5) = 31.
		

Crossrefs

Cf. A000122, A320932, A321181, A321186.

Formula

a(n) = [x^(n*(n+1)*(2*n+1)/6)] Product_{k=1..n} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function.