A188569 -id:A188569 - cp's OEIS frontend

A220515 Numbers n such that A183054(n) is not equal to A188569(n).

24, 47, 49, 74, 96, 99, 116, 124, 145, 149, 162, 174, 194, 199, 224, 237, 243, 249, 274, 277, 292, 299, 324, 331, 341, 346, 349, 358, 374, 390, 399, 424, 439, 449, 474, 479, 488, 499, 500, 507, 524, 537, 549, 566, 574, 586, 599, 600, 624, 635

Offset: 1

Omar E. Pol, Feb 27 2013

nonn

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. For more information see A222031.

			First three terms are 24, 47, 49 because first 50 terms of A183054 coincide with first 50 terms of A188569 except for the indices 24, 47, 49 as shown below:
(A183054(24) = 3) < (A188569(24) = 21).
(A183054(47) = 3) < (A188569(47) = 27).
(A183054(49) = 5) < (A188569(49) = 35).
Observation:
A183054(24) = A188569(24)/7 = 3.
A183054(47) = A188569(47)/9 = 3.
A183054(49) = A188569(49)/7 = 5.

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

Cf. A000041, A183010, A183011, A183054, A188569, A222031, A222032.

a(4)-a(50) from Giovanni Resta, Mar 04 2013

A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

-1, 0, 0, 4, 6, 0, 0, 12, 12, 0, 0, 12, 16, 0, 0, 24, 18, 0, 0, 12, 24, 0, 0, 36, 24, 0, 0, 16, 24, 0, 0, 36, 36, 0, 0, 24, 30, 0, 0, 48, 24, 0, 0, 12, 48, 0, 0, 60, 40, 0, 0, 24, 24, 0, 0, 48, 48, 0, 0, 36, 48, 0, 0, 60, 42, 0, 0, 12, 48, 0, 0, 84, 36, 0, 0

Offset: 0

Michael Somos, Jul 05 2015

sign

Coefficients of q-expansion of Eisenstein series G_{3/2}(tau) multiplied by 12. - N. J. A. Sloane, Mar 16 2019

			G.f. = -1 + 4*x^3 + 6*x^4 + 12*x^7 + 12*x^8 + 12*x^11 + 16*x^12 + 24*x^15 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Kathrin Bringmann and Jeremy Lovejoy, Overpartitions and class numbers of binary quadratic forms, arXiv:0712.0631 [math.NT], 2007. See page 5, equation (1.12).
Archer Clayton, Helen Dai, Tianyu Ni, Erick Ross, Hui Xue, and Jake Zummo, Non-repetition of second coefficients of Hecke polynomials, arXiv:2411.18419 [math.NT], 2024. See p. 21.
Wikipedia, Hurwitz class number.
Don Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A004024, A005869, A005876, A005877, A005878, A005884, A005886, A008443.
Cf. A045828, A045831, A045834, A058305, A058306, A130695, A181648, A185220.
Cf. A188569, A213022, A213617, A213624, A213625, A213627, A259827.
See also A306934-A306937.

terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2 k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; gf[terms // Sqrt // Ceiling] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2017 *)
a[ n_] := If[ n<1, -Boole[n==0], With[{m = Floor[(-1 + Sqrt[1 + 4*n])/2]}, -2*SeriesCoefficient[ Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, -m-1,m}] / EllipticTheta[3, 0, x] + Sum[(-1)^k*x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, -m-2,m}]/ EllipticTheta[3, 0, -x], {x, 0, n}]]]; (* Michael Somos, Feb 04 2022 *)

PARI
```
{a(n) = 12 * qfbhclassno(n)};
```

{a(n) = my(D, f); 12 * if( n<1, (n==0)/-12, [D, f] = core(-n, 1); if( D%4>1 && !(f%2), D*=4; f/=2); if( D%4<2, qfbclassno(D) / max(1, D+6), 0) * sumdiv(f, d, moebius(d) * kronecker(D, d) * sigma(f/d)))};

a(n) = 12 * A058305(n) / A058306(n). a(4*n + 1) = a(4*n + 2) = 0. a(3*n + 4) = 6 * A259827(n).
a(4*n + 3) = 4 * A130695(n). a(8*n + 3) = A005886(n) = 2 * A005869(n) = 4 * A008443(n). a(12*n + 7) = 12 * A259655(n).
a(16*n + 4) = 6 * A045834(n) = 3 * A005876(n). a(16*n + 8) = 12 * A045828(n) = 6 * A005884(n) = 3 * A005877(n).
a(24*n + 3) = 4 * A213627(n). a(24*n + 7) = 12 * A185220(n). a(24*n + 11) = 12 * A213617(n). a(24*n + 19) = 12 * A181648(n). a(24*n + 23) = 12 * A188569(n+1).
a(32*n + 4) = 6 * A213022(n). a(32*n + 8) = 12 * A213625(n). a(32*n + 12) = 16 * A008443(n) = 8 * A005869(n) = 4 * A005886(n) = 2 * A005878(n). a(32*n + 20) = 24 * A045831(n) = 6 * A004024(n). a(32*n + 24) = 24 * A213624(n).
G.f.: -2 * (Sum_{k in Z} (-1)^k * x^(k*k + k) / (1 + (-x)^k)^2) / (Sum_{k in Z} x^k^2) - 2 * (Sum_{k in Z} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (Sum_{k in Z} (-x)^k^2).
a(n) >= 0 if n > 0. - Michael Somos, Feb 04 2022

A183054 h(-24*n+1) where h(d) is the class number of the quadratic field with discriminant d.

Original entry on oeis.org

3, 5, 7, 8, 10, 10, 11, 13, 14, 15, 13, 14, 19, 18, 19, 17, 16, 21, 20, 25, 21, 18, 26, 3, 25, 22, 23, 30, 24, 31, 21, 22, 32, 30, 33, 21, 29, 31, 28, 36, 27, 30, 35, 36, 34, 23, 3, 41, 5, 38, 35, 26, 40, 36, 45, 34, 25, 44, 34, 39, 32, 37, 49, 38, 51, 33

Offset: 1

Omar E. Pol, Jul 14 2011

nonn

First differs from A188569 at a(24). It appears that this coincides with A188569 in a large number of terms. - Omar E. Pol, Feb 20 2013

Cf. A188569.

Table[NumberFieldClassNumber[Sqrt[-24*n + 1]], {n, 100}] (* T. D. Noe, Jul 15 2011 *)

More terms and new definition suggested by David Scambler, Jul 15 2011
Extended by T. D. Noe, Jul 15 2011
Edited by Omar E. Pol, Feb 26 2013

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.

Original entry on oeis.org

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.

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A220515 Numbers n such that A183054(n) is not equal to A188569(n).

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A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

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A183054 h(-24*n+1) where h(d) is the class number of the quadratic field with discriminant d.

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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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Formula

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.

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