A186924 -id:A186924 - cp's OEIS frontend

A058487 McKay-Thompson series of class 12I for the Monster group.

1, 2, 1, 0, -2, -2, 2, 4, 3, -4, -8, -4, 5, 14, 7, -8, -20, -12, 14, 28, 17, -20, -44, -24, 28, 66, 36, -40, -90, -52, 56, 124, 71, -80, -176, -96, 109, 244, 133, -144, -326, -182, 198, 432, 240, -268, -580, -316, 349, 772, 420, -456, -1004, -552, 600, 1300, 713, -780, -1692, -916, 1001, 2186, 1182

Offset: 0

N. J. A. Sloane, Nov 27 2000

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 5 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(12). [Yang 2004] - Michael Somos, Jul 21 2014

			G.f. = 1 + 2*x + x^2 - 2*x^4 - 2*x^5 + 2*x^6 + 4*x^7 + 3*x^8 - 4*x^9 - 8*x^10 + ...
T12I = 1/q + 2*q + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..502 from G. A. Edgar)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, A062243, A128633, A186924, A217786.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / EllipticTheta[ 2, 0, q^3]^2, {q, 0, 2 n - 1}]; (* Michael Somos, Jul 21 2014 *)
QP = QPochhammer; s = QP[q^2]^4*(QP[q^3]^2/(QP[q]^2*QP[q^6]^4)) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

{a(n) = local(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 * (A + 3*x) / (A - x))); polcoeff(A, n))};

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2))^2, n))};

Expansion of (psi(x) / psi(x^3))^2 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 21 2014
G.f.: ( Product_{k>0} (1 - x^(6*k - 2)) * (1 - x^(6*k - 4)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k - 5))) )^2.
Expansion of q^(1/2) * (eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^4)) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, 0,...]. - Michael Somos, Mar 18 2004
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + 3*v - u^2*v + v^2. - Michael Somos, Mar 18 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A186924.
a(n) = (-1)^n * A062243(n).
Convolution square is A128633. Convolution inverse of A217786. - Michael Somos, Jul 21 2014

A187100 Expansion of q * (psi(-q^3) * psi(q^6)) / (psi(-q) * phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 7, 15, 30, 57, 104, 183, 313, 522, 852, 1365, 2150, 3336, 5106, 7719, 11538, 17067, 25004, 36306, 52280, 74700, 105960, 149277, 208951, 290706, 402127, 553224, 757158, 1031166, 1397744, 1886151, 2534316, 3391254, 4520112, 6002007, 7940846

Offset: 1

Michael Somos, Mar 05 2011

nonn

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

			q + 3*q^2 + 7*q^3 + 15*q^4 + 30*q^5 + 57*q^6 + 104*q^7 + 183*q^8 + 313*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Johannes Blümlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A186924, A187130, A121373, A000122, A010054, A000700.

nmax = 50; CoefficientList[Series[Product[(1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^3 / ((1-x^k)^3 * (1-x^(4*k)) * (1-x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[n_] := SeriesCoefficient[(EllipticTheta[2, 0, I*q^(3/2)]* EllipticTheta[2, 0, q^3])/(2*EllipticTheta[2, 0, I*q^(1/2)]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[a[n], {n, 50}] (* G. C. Greubel, Nov 27 2017 *)
eta[q_] := q^(1/24)*QPochhammer[q]; A:= eta[q^2]^2*eta[q^3]*eta[q^12]^3/ (eta[q]^3*eta[q^4]*eta[q^6]^2); a:=CoefficientList[Series[A/q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^6 + A)^2), n))}

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^12)^3 / (eta(q)^3 * eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/12) * 1/f(t) where q = exp(2 Pi i t).
Convolution inverse of A187130. A186924(n) = 4 * a(n) unless n=0.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 12, -28, 60, -120, 228, -416, 732, -1252, 2088, -3408, 5460, -8600, 13344, -20424, 30876, -46152, 68268, -100016, 145224, -209120, 298800, -423840, 597108, -835804, 1162824, -1608508, 2212896, -3028632, 4124664, -5590976, 7544604, -10137264, 13565016

Offset: 0

Michael Somos, Aug 14 2015

sign

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

			G.f. = 1 - 4*x + 12*x^2 - 28*x^3 + 60*x^4 - 120*x^5 + 228*x^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A132002, A186924, A260215, A233673.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10 / (eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};

Expansion of eta(q)^4 * eta(q^4)^4 * eta(q^6)^10 / ( eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -4, 6, 0, 2, -4, 0, -4, 2, 0, 6, -4, 0, ...].
G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2)^2.
G.f.: (Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)))^2.
a(n) = (-1)^n * A186924(n) = A233673(3*n) = A260215(3*n).
Convolution square of A132002.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, 4, -12, 8, 12, -32, 20, 28, -72, 48, 60, -152, 96, 120, -300, 184, 228, -560, 344, 416, -1008, 608, 732, -1756, 1048, 1252, -2976, 1768, 2088, -4928, 2900, 3408, -7992, 4672, 5460, -12728, 7408, 8600, -19944, 11544, 13344, -30800, 17744, 20424

Offset: 0

Michael Somos, Mar 24 2013

sign

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

			G.f. = 1 - 4*x + 4*x^2 + 4*x^3 - 12*x^4 + 8*x^5 + 12*x^6 - 32*x^7 + 20*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A123649, A186924, A217786.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Mar 24 2013 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^4), n))}

Expansion of eta(q)^4 * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)^4) in powers of q.
Euler transform of period 6 sequence [ -4, -2, 0, -2, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u) * (u + v^2) - 4 * u.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (3 + u * v)^2 - v * (3*u + v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217786.
a(n) = - 4 * A123649(n) unless n=0.
Convolution inverse of A186924. Convolution square of A139137.

A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 4, 6, 8, 12, 16, 22, 28, 36, 48, 60, 76, 96, 120, 150, 184, 228, 280, 340, 416, 504, 608, 732, 878, 1052, 1252, 1488, 1768, 2088, 2464, 2902, 3408, 3996, 4672, 5460, 6364, 7400, 8600, 9972, 11544, 13344, 15400, 17752, 20424, 23472, 26944, 30876, 35346

Offset: 0

Michael Somos, Aug 10 2015

nonn

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

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 22*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A123629, A186924, A187100, A212484, A233693, A260215.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ -q, q] QPochhammer[ q^9, q^18] QPochhammer[ q^9, -q^9], {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3), n))};

Expansion of eta(q^2)^3 * eta(q^9)^2 * eta(q^36) / (eta(q)^2 * eta(q^4) * eta(q^18)^3) in powers of q.
G.f. A(x) = B(x) / B(x^9) where B(x) is the g.f. of A080054.
Euler transform of period 36 sequence [ 2, -1, 2, 0, 2, -1, 2, 0, 0, -1, 2, 0, 2, -1, 2, 0, 2, 0, 2, 0, 2, -1, 2, 0, 2, -1, 0, 0, 2, -1, 2, 0, 2, -1, 2, 0, ...].
a(n) = 2 * A233693(n) unless n=0.  a(2*n) = 2 * A123629(n) = 2 * A212484(n) unless n=0.
a(3*n) = A186924(n). a(3*n) = 4 * A187100(n) unless n=0.
a(n) = (-1)^n * A260215(n). - Michael Somos, Aug 14 2015
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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A058487 McKay-Thompson series of class 12I for the Monster group.

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A187100 Expansion of q * (psi(-q^3) * psi(q^6)) / (psi(-q) * phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

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A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

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A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.

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A261156 Expansion of chi(q) * chi(-q^9) / (chi(-q) * chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

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