A022055 -id:A022055 - cp's OEIS frontend

A055749 Jacobi form of weight 12 and index 1 for the Niemeier lattice of type D_24.

1, 0, 0, 88, 926, 0, 0, 24640, 118932, 0, 0, 3786312, 9240664, 0, 0, 94951872, 187974558, 0, 0, 1145140744, 1960908360, 0, 0, 8505386304, 13288248072, 0, 0, 45751234016, 67083722816, 0, 0, 195423783552, 272571256980, 0, 0

Offset: 0

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

nonn

Eichler and Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Cf. A022055, A055747, A003785.

E_8*E_{4, 1}+32*phi_12. phi_12=unique Jacobi cusp form of weight 12 index 1.

G.f.: b(z) + 32*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022

A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 4, 2, 0, 1, 12, 4, 0, 0, 1, 24, 6, 0, 0, 0, 1, 40, 24, 24, 4, 0, 0, 1, 60, 90, 96, 12, 8, 0, 0, 1, 84, 252, 240, 24, 24, 0, 0, 0, 1, 112, 574, 544, 200, 144, 8, 0, 2, 0, 1, 144, 1136, 1288, 1020, 560, 96, 48, 4, 0, 0, 1, 180, 2034, 3136, 3444, 1560, 400, 192, 6, 4, 0, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			Square array begins:
1,  1,  1,   1,    1,    1,  ...
0,  0,  4,  12,   24,   40,  ...
0,  2,  4,   6,   24,   90,  ...
0,  0,  0,  24,   96,  240,  ...
0,  0,  4,  12,   24,  200,  ...
0,  0,  8,  24,  144,  560,  ...

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 118.

Columns k=0..32 give A000007, A089798 (absolute values), A004018, A004015, A004011, A005930, A008428, A008429, A008430, A008431, A008432, A022042, A022043, A022044, A022045, A022046, A022047, A022048, A022049, A022050, A022051, A022052, A022053, A022054, A022055, A022056, A022057, A022058, A022059, A022060, A022061, A022062, A022063.
Main diagonal gives A303333.
Cf. A122141, A286815.

Table[Function[k, SeriesCoefficient[(EllipticTheta[3, 0, q^(1/2)]^k + EllipticTheta[4, 0, q^(1/2)]^k)/2, {q, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2, where theta_() is the Jacobi theta function.

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A055749 Jacobi form of weight 12 and index 1 for the Niemeier lattice of type D_24.

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A297331 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (theta_3(q^(1/2))^k + theta_4(q^(1/2))^k)/2.

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