A029840 -id:A029840 - cp's OEIS frontend

A029839 McKay-Thompson series of class 16B for the Monster group.

1, 2, -1, -2, 3, 2, -4, -4, 5, 8, -8, -10, 11, 12, -15, -18, 22, 26, -29, -34, 38, 42, -51, -56, 66, 78, -85, -98, 109, 120, -139, -156, 176, 202, -222, -250, 279, 306, -346, -384, 429, 482, -530, -590, 650, 714, -797, -876, 972, 1080, -1180, -1304, 1431, 1562, -1728, -1892, 2078, 2290, -2496

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

In [Klein and Fricke 1890], the g.f. A(q)/2 is denoted by mu. On page 613 special values given are mu(i infinity) = infinity, mu(0) = 1, mu(2) = -1 and on page 615 properties given are mu(omega+1) = -i mu(omega), mu(-1/omega) = (mu(omega)+1)/(mu(omega)-1). - Michael Somos, Nov 09 2014

			G.f. = 1 + 2*x - x^2 - 2*x^3 + 3*x^4 + 2*x^5 - 4*x^6 - 4*x^7 + 5*x^8 + 8*x^9 + ...
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
F. Klein and R. Fricke, Vorlesungen über die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1890, Vol. 1, see pp. 613, 615, 675.
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group

Cf. A079006, A082304, A029838.

Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^b: this sequence (b=1), A029839 (b=2), A029840 (b=3), A029841 (b=4), A029842 (b=5), A029843 (b=6), A029844 (b=7).

a[0] = 1; a[n_] := Module[{A, m}, If[n < 0, 0, A = 1; m = 1; While[m <= n, m *= 2; A = A /. x -> x^2; A = Sqrt[A + 4*x/A]]; SeriesCoefficient[A, {x, 0, n}]]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Mar 12 2014, after PARI *)
a[ n_] := SeriesCoefficient[ 2 q^(1/4) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + 4*x/A)); polcoeff(A, n))};

Expansion of q times normalized Hauptmodul for Gamma(4) in powers of q^4.
Expansion of q^(1/4) * eta(q^2)^6 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [2, -4, 2, 0, ...].
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x / A(x^2). - Michael Somos, Mar 08 2004
G.f.: Product_{k>0} ((1 + x^(2*k-1)) / (1 + x^(2*k)))^2.
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^2 - u^2*v. - Michael Somos, May 14 2004
Given g.f. A(x), then B(q) = A(q^4) / (2*q) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Oct 04 2006
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u6 + u2)^2 - u1*u2*u3*u6. - Michael Somos, Oct 04 2006
Convolution inverse of A079006.
Expansion of q^(1/4) * 2 / k(q)^(1/2) in powers of Jacobi nome q where k() is the elliptic modulus.
Expansion of q^(1/2) * 2 * (1 + k'(q)) / k(q) in powers of q^2. - Michael Somos, Nov 09 2014
Expansion of phi(x) / psi(x^2) = phi(x)^2 / psi(x)^2 = psi(x)^2 / psi(x^2)^2 = phi(-x^2)^2 / psi(-x)^2 = chi(-x^2)^4 / chi(-x)^2 = chi(x)^2 * chi(-x^2)^2 = chi(x)^4 * chi(-x)^2 = f(x)^2 / f(-x^4)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of continued fraction 1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)) in powers of x^4. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007096.
a(n) = (-1)^n * A082304(n). Convolution square is A029841. - Michael Somos, Jul 05 2014
From Peter Bala, Jan 09 2021: (Start)
A(q) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+1)) / Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)).
A(q) = ( 1 + q/(1 + (q + q^2)/(1 + q^3/(1 + (q^2 + q^4)/(1 + q^5/(1 + ... ))))) )^2. See Agarwal, p. 285.
A(q) = B(q)^2, where B(q) is the g.f. of A029838. (End)
abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

Additional comments from Michael Somos, Jul 11 2002

A029842 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^5.

Original entry on oeis.org

1, 5, 5, -10, -10, 31, 20, -75, -40, 150, 84, -280, -165, 520, 290, -935, -495, 1595, 855, -2640, -1424, 4315, 2265, -6925, -3570, 10860, 5605, -16740, -8615, 25520, 12984, -38455, -19390, 57150, 28740, -83961, -42110, 122320

Offset: 0

N. J. A. Sloane

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A029838, A029839, A029840, A029841, A029843, A029844.

G.f.: x^(5/8)*theta_2(sqrt(x))^5/theta_2(x)^5, where theta_() is the Jacobi theta function. - Ilya Gutkovskiy, Dec 04 2017

A029843 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^6.

Original entry on oeis.org

1, 6, 9, -10, -24, 36, 65, -102, -153, 232, 327, -468, -663, 918, 1287, -1768, -2391, 3240, 4289, -5676, -7488, 9758, 12753, -16524, -21250, 27300, 34758, -44128, -55896, 70380, 88519, -110874, -138285, 172136, 213315

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A029838, A029839, A029840, A029841, A029842, A029844.

G.f.: x^(3/4)*theta_2(sqrt(x))^6/theta_2(x)^6, where theta_() is the Jacobi theta function. - Ilya Gutkovskiy, Dec 04 2017

A029844 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^7.

Original entry on oeis.org

1, 7, 14, -7, -42, 28, 133, -90, -357, 231, 833, -511, -1792, 1064, 3695, -2163, -7329, 4221, 13923, -7847, -25536, 14161, 45703, -25109, -80010, 43526, 136941, -73654, -229823, 122493, 379582, -200935, -617729, 324751

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A029838, A029839, A029840, A029841, A029842, A029843.

G.f.: x^(7/8)*theta_2(sqrt(x))^7/theta_2(x)^7, where theta_() is the Jacobi theta function. - Ilya Gutkovskiy, Dec 04 2017

A187053 Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 9, -22, 48, -99, 194, -363, 657, -1155, 1977, -3312, 5443, -8787, 13968, -21894, 33873, -51795, 78345, -117312, 174033, -255945, 373353, -540486, 776848, -1109040, 1573209, -2218198, 3109713, -4335840, 6014123, -8300811, 11402928

Offset: 0

Michael Somos, Mar 06 2011

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 3*x + 9*x^2 - 22*x^3 + 48*x^4 - 99*x^5 + 194*x^6 - 363*x^7 + ...
G.f. = q^3 - 3*q^11 + 9*q^19 - 22*q^27 + 48*q^35 - 99*q^43 + 194*q^51 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001937, A029840, A083365.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^3, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^3, n))};

Expansion of q^(-3/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^3 in powers of q.
Euler transform of period 4 sequence [-3, 6, -3, 0, ...].
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^3.
Convolution inverse of A029840. Convolution cube of A083365. a(n) = (-1)^n * A001937(n).
a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A296043 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^n.

Original entry on oeis.org

1, 1, -1, -5, -1, 31, 65, -90, -641, -644, 3329, 11386, -1471, -87021, -164634, 317935, 1881471, 1418719, -11370760, -33937951, 17468929, 294971868, 468897758, -1304743033, -6275603903, -2804572819, 42665919997, 109181454826, -106020803386, -1063546684834, -1362993953395

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

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G. C. Greubel, Table of n, a(n) for n = 0..500

Main diagonal of A296067.

Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A296044, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^n, {x, 0, n}], {n, 0, 30}]

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, -1, 0, 1, 3, -1, 0, 0, 1, 4, 0, -2, 1, 0, 1, 5, 2, -5, 3, 0, 0, 1, 6, 5, -8, 3, 2, -1, 0, 1, 7, 9, -10, -1, 9, -4, -1, 0, 1, 8, 14, -10, -10, 20, -7, -4, 2, 0, 1, 9, 20, -7, -24, 31, -2, -15, 5, 1, 0, 1, 10, 27, 0, -42, 36, 20, -40, 9, 8, -2, 0, 1, 11, 35, 12, -62, 28, 65, -75, 3, 27, -8, -1, 0

Offset: 0

Ilya Gutkovskiy, Dec 04 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k - 3)*x^2 + (1/6)*k*(k^2 - 9*k + 8)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 59*k - 18)*x^4 + (1/120)*k*(k^4 - 30*k^3 + 215*k^2 - 330*k + 144)*x^5 + ...
Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0, -1, -1,  0,   2,   5,  ...
0,  0, -2, -5,  -8, -10,  ...
0,  1,  3,  3,  -1, -10,  ...
0,  0,  2,  9,  20,  31,  ...

G. C. Greubel, Rows n=0..100 of antidiagonals, flattened

Columns k=0..8 give A000007, A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845 (with offset 0).
Main diagonal gives A296043.
Cf. A296068.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i - 1))/(1 + x^(2 i)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

G.f. of column k: (x^(1/8)*theta_2(sqrt(x))/theta_2(x))^k, where theta_() is the Jacobi theta function.

cp's OEIS Frontend

A029839 McKay-Thompson series of class 16B for the Monster group.

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A029842 Expansion of Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^5.

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A029843 Expansion of Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^6.

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A029844 Expansion of Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^7.

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A187053 Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.

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A296043 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^n.

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A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

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A296047 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^k.

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A029842 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^5.

A029843 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^6.

A029844 Expansion of Product_{m>=1} ((1+q^(2m-1))/(1+q^(2m)))^7.

A296043 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^n.

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.