A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, ... 1, 3, 6, 11, 20, 36, ... 1, 4, 11, 29, 70, 157, ... 1, 5, 17, 54, 165, 482, ... 1, 6, 26, 99, 357, 1203, ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1274
- Eric Weisstein's World of Mathematics, Jacobi Theta Functions
- Index entries for sequences related to sums of squares
Crossrefs
Programs
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Mathematica
Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten -
PARI
T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019
Formula
A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.
Comments