A306518 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{d|k} theta_3(q^d).

1, 1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 4, 2, 1, 2, 2, 2, 2, 0, 1, 2, 0, 4, 6, 0, 0, 1, 2, 2, 0, 4, 0, 4, 0, 1, 2, 0, 6, 2, 4, 0, 0, 0, 1, 2, 2, 0, 6, 2, 8, 4, 2, 2, 1, 2, 0, 4, 2, 4, 4, 8, 0, 6, 0, 1, 2, 2, 2, 4, 0, 14, 0, 6, 2, 0, 0, 1, 2, 0, 4, 6, 4, 0, 8, 0, 6, 0, 4, 0, 1, 2, 2, 0, 2, 0, 8, 2, 6, 6, 8, 0, 4, 0

Offset

0,3

Links

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

G.f. of column k: Product_{d|k} theta_3(q^d).

Example

Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  0,  2,  0,  2,  0,  2,  ...
  0,  4,  2,  4,  0,  6,  ...
  2,  2,  6,  4,  2,  6,  ...
  0,  0,  0,  4,  2,  4,  ...

Mathematica

Table[Function[k, SeriesCoefficient[Product[EllipticTheta[3, 0, q^d], {d, Divisors[k]}], {q, 0, n}]][i - n + 1], {i, 0, 13}, {n, 0, i}] // Flatten

Crossrefs

Columns k=1..48 give A000122, A033715, A033716, A033717, A033718, A033712, A033719, A033720, A033721, A033722, A033723, A033724, A033725, A033726, A033727, A033728, A033729, A033730, A033731, A033732, A033733, A033734, A033735, A033736, A033737, A033738, A033739, A033740, A033741, A033742, A033743, A033744, A033745, A033746, A033747, A033748, A033749, A033750, A033751, A033752, A033753, A033754, A033755, A033756, A033757, A033758, A033759, A033760.

Cf. A320305 (diagonal).

Sequence in context: A115628 A114002 A114004 * A333310 A049986 A218797

Adjacent sequences: A306515 A306516 A306517 * A306519 A306520 A306521

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Feb 21 2019

Status

approved