A220515 -id:A220515 - cp's OEIS frontend

A222031 Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

1, -23, 3592, -419, 1, -94, 169659, -65838, 1092873176, 145023, 1, -213, 1312544, -723721, 44648582886, 9188934683, 166629520876208, 2791651635293, 1, -475, 9032603, -9455070, 3949512899743, -97215753021, 9776785708507683, -53144327916296, -134884469547631

Offset: 1

Omar E. Pol, Mar 04 2013

For an algorithm to compute the partition class polynomial Hpart_n(x) see the Bruinier-Ono-Sutherland paper, 3.3. Algorithm 3, p. 15-19.

Note that the absolute value of T(n,2) is also the trace Tr(n) = A183011(n), the numerator of the finite algebraic formula for the number of partitions of n. The formula is p(n) = Tr(n)/(24*n - 1). See theorem 1.1 in the Bruinier-Ono paper.

			For n = 1 the first partition class polynomial Hpart_1(x) is x^3 - 23*x^2 + 3592/23*x - 419, so the numerators of the coefficients are 1, -23, 3592, -419.
Triangle begins:
1, -23, 3592, -419;
1, -94, 169659, -65838, 1092873176, 145023;
1, -213, 1312544, -723721, 44648582886, 9188934683, 166629520876208, 2791651635293;
1, -475, 9032603, -9455070, 3949512899743, -97215753021, 9776785708507683, -53144327916296, -134884469547631;
...

J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
J. H. Bruinier, K. Ono, A. V. Sutherland, Class polynomials for nonholomorphic modular functions
A. V. Sutherland, Partition class polynomials, Hpart_n(x), for n = 1..770

Row n has length 1 + A188569(n). Absolute values of column 2 give A183011. Columns 3-4: A183007, A187218. For denominators see A222032.

Cf. A000041, A183010, A220515.

abs(T(n,2))/(24n-1) = A183011(n)/A183010(n) = A000041(n).

A222032 Irregular triangle read by rows in which row n gives denominators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

Original entry on oeis.org

1, 1, 23, 1, 1, 1, 47, 1, 2209, 47, 1, 1, 71, 1, 5041, 71, 357911, 5041, 1, 1, 95, 1, 9025, 19, 857375, 361, 11875, 1, 1, 119, 1, 2023, 1, 240737, 14161, 200533921, 1685159, 4857223, 1, 1, 143, 1, 1573, 143, 2924207, 20449, 418161601, 2924207, 27217619

Offset: 1

Omar E. Pol, Mar 04 2013

For more information see A222031.

			For n = 1 the first partition class polynomial Hpart_1(x) is x^3 - 23*x^2 + 3592/23*x - 419, so the denominators of the coefficients are 1, 1, 23, 1.
Triangle begins:
1, 1, 23, 1;
1, 1, 47, 1, 2209, 47;
1, 1, 71, 1, 5041, 71, 357911, 5041;
1, 1, 95, 1, 9025, 19, 857375, 361, 11875;
1, 1, 119, 1, 2023, 1, 240737, 14161, 200533921, 1685159, 4857223;
1, 1, 143, 1, 1573, 143, 2924207, 20449, 418161601, 2924207, 27217619;
1, 1, 167, 1, 27889, 167, 4657463, 27889, 777796321, 4657463, 129891985607, 777796321;

A. V. Sutherland, Partition class polynomials, Hpart_n(x), for n = 1..770

Row n has length 1 + A188569(n). For numerators see A222031.

Cf. A000041, A183007, A183010, A183011, A187218, A188569, A220515.

cp's OEIS Frontend

A222031 Irregular triangle read by rows in which row n gives numerators of the coefficients of the partition class polynomial Hpart_n(x), n >= 1.

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