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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A321183 a(n) = [x^((n*(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k^2*m).

Original entry on oeis.org

1, 1, 3, 26, 438, 11674, 434613, 21040885, 1263748763, 91057116368, 7676892453542, 742890018054927, 81267790173334794, 9926903213704358577, 1340280764681712515084, 198320073897808037293388, 31929177807445245255119558, 5558580993355817894674501169, 1040777481846356463369367882750
Offset: 0

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Author

Seiichi Manyama, Oct 29 2018

Keywords

Comments

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation 1^2*a_1 + 2^2*a_2 + ... + n^2*a_n = (n*(n+1)/2)^2.

Examples

			1^2* 0 + 2^2*0 + 3^2*4 = 36.
1^2* 0 + 2^2*9 + 3^2*0 = 36.
1^2* 1 + 2^2*2 + 3^2*3 = 36.
1^2* 2 + 2^2*4 + 3^2*2 = 36.
1^2* 3 + 2^2*6 + 3^2*1 = 36.
1^2* 4 + 2^2*8 + 3^2*0 = 36.
1^2* 5 + 2^2*1 + 3^2*3 = 36.
1^2* 6 + 2^2*3 + 3^2*2 = 36.
1^2* 7 + 2^2*5 + 3^2*1 = 36.
1^2* 8 + 2^2*7 + 3^2*0 = 36.
1^2* 9 + 2^2*0 + 3^2*3 = 36.
1^2*10 + 2^2*2 + 3^2*2 = 36.
1^2*11 + 2^2*4 + 3^2*1 = 36.
1^2*12 + 2^2*6 + 3^2*0 = 36.
1^2*14 + 2^2*1 + 3^2*2 = 36.
1^2*15 + 2^2*3 + 3^2*1 = 36.
1^2*16 + 2^2*5 + 3^2*0 = 36.
1^2*18 + 2^2*0 + 3^2*2 = 36.
1^2*19 + 2^2*2 + 3^2*1 = 36.
1^2*20 + 2^2*4 + 3^2*0 = 36.
1^2*23 + 2^2*1 + 3^2*1 = 36.
1^2*24 + 2^2*3 + 3^2*0 = 36.
1^2*27 + 2^2*0 + 3^2*1 = 36.
1^2*28 + 2^2*2 + 3^2*0 = 36.
1^2*32 + 2^2*1 + 3^2*0 = 36.
1^2*36 + 2^2*0 + 3^2*0 = 36.
So a(3) = 26.

Crossrefs

Cf. A000537, A037444, A321181, A321186.

Extensions

a(16)-a(18) from Alois P. Heinz, Oct 29 2018

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Oct 30 2018

Keywords

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.
Showing 1-2 of 2 results.

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