A132972 -id:A132972 - cp's OEIS frontend

A062244 McKay-Thompson series of class 36B for the Monster group.

1, -1, 1, 1, -1, 0, 1, -2, 0, 2, -3, 1, 4, -4, 1, 4, -6, 1, 5, -8, 1, 8, -10, 2, 11, -14, 4, 14, -19, 4, 17, -24, 4, 23, -31, 6, 31, -40, 9, 38, -50, 10, 46, -63, 11, 60, -79, 16, 77, -98, 21, 92, -122, 24, 112, -150, 28, 140, -183, 36, 173, -224, 46, 208, -273, 54, 249, -329, 62, 304, -396, 78, 370, -478, 98

Offset: 0

N. J. A. Sloane, Jul 01 2001

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

			G.f. = 1 - x + x^2 + x^3 - x^4 + x^6 - 2*x^7 + 2*x^9 - 3*x^10 + x^11 + ...
T36B = 1/q - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group

Cf. A062242, A092848, A128111, A132179, A132972.

a[ n_] := (-1)^n SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* Michael Somos, Aug 20 2014 *)
A062244[n_] := SeriesCoefficient[QPochhammer[-q^3, q^6]^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A062244[n], {n,0,50}] (* G. C. Greubel, Aug 09 2017 *)
a[ n_] := SeriesCoefficient[ 2 x^(1/3) / (EllipticTheta[ 3, 0, x^(1/3)] / EllipticTheta[ 3, 0, x^3] - 1), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)
a[ n_] := SeriesCoefficient[ x^(1/3) (1 + EllipticTheta[2, Pi/4, x^(1/6)] / EllipticTheta[2, Pi/4, x^(3/2)]), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^3]/(QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 10 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))}; /* Michael Somos, Jan 09 2005 */

Expansion of a Hauptmodul for Gamma'_0(18).
G.f.: Product_{k>0} (1 + x^(6*k - 3))^3 / (1 + x^(2*k - 1)). - Michael Somos, Mar 17 2004
Expansion of q^(1/3) * eta(q) * eta(q^4) * eta(q^6)^6 / (eta(q^2)^2 * eta(q^3)^3 * eta(q^12)^3) in powers of q.
Given g.f. A(x), then B(q) = A(q^3) /q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 - u^3*v^3 - 3*u^2*v^2 + v*u^4 + 4*u*v - 2. - Michael Somos, Mar 17 2004
Expansion of chi(x^3)^3 / chi(x) = f(-x, x^2) / psi(-x^3) = phi(x^3) / f(x, x^5) in powers of x where phi(), psi(), chi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Aug 10 2017
Euler transform of period 12 sequence [-1, 1, 2, 0, -1, -2, -1, 0, 2, 1, -1, 0, ...]. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132972.
a(n) = (-1)^n * A062242(n). a(2*n) = A132179(n). a(2*n + 1) = - A092848(n).
Convolution inverse of A128111.

A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 21, 48, 99, 198, 375, 693, 1236, 2160, 3681, 6168, 10140, 16434, 26235, 41376, 64449, 99342, 151530, 229032, 343068, 509760, 751509, 1099998, 1598925, 2309274, 3314541, 4729920, 6711993, 9474624, 13306506, 18598437, 25874460, 35838288, 49427640, 67892592

Offset: 0

Seiichi Manyama, Nov 07 2016

nonn

			G.f.: 1 + 3*x + 9*x^2 + 21*x^3 + 48*x^4 + 99*x^5 + 198*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), this sequence (k=3), A274327 (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A005928, A132972.

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^3, x^3]/QPochhammer[x, x]^3 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q)^3)} \\ Altug Alkan, Mar 20 2018

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016
G.f.: (x^3; x^3)inf/((x; x)_inf)^3, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017
It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - x^(3*n + 2))) ). Cf. A132972.  - Peter Bala, Dec 23 2021

A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548

Offset: 1

Michael Somos, Sep 07 2007

nonn

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

			G.f. = q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10001
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S115.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A128129, A128640, A132302, A132972, A132976. Essentially the same as A213267.

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) * (1+x^(18*k)) / (1-x^(4*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)), n))};

Expansion of eta(q^2) * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * u2 - (1 + u1 + u2) * (u3 + u6 + 3 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132976.
G.f.: x * Product_{k>0} P(3,x^k) * P(9,x^k) * P(12,x^k) * P(36,x^k) where P(n,x) is the n-th cyclotomic polynomial.
3 * a(n) = A132972(n) unless n=0. a(2*n) = A128129(n). a(2*n + 1) = A132302(n). a(3*n) = A128640(n). Convolution inverse of A132976.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 3, -3, 6, -9, 12, -15, 21, -30, 36, -45, 60, -78, 96, -117, 150, -189, 228, -276, 342, -420, 504, -603, 732, -885, 1050, -1245, 1488, -1773, 2088, -2454, 2901, -3420, 3996, -4662, 5460, -6378, 7404, -8583, 9972, -11565, 13344, -15378, 17748, -20448

Offset: 0

Michael Somos, Jun 04 2008, Aug 12 2009

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
For n >= 1, a(n)/3 is a weighted count of overpartitions with restricted odd differences. Namely, the number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is odd and overlined. - Jeremy Lovejoy, Aug 07 2015

			G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 6*q^4 - 9*q^5 + 12*q^6 - 15*q^7 + 21*q^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A092848, A124243, A128128, A132302.

with(numtheory):
a:= proc(n) option remember:
      `if`(n=0, 1, add(add(d*[0, -3, 0, -2, 0, -3]
      [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 08 2015

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^3 QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d{0, -3, 0, -2, 0, -3}[[Mod[d, 6]+1]], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

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

Expansion of chi(-q)^3 / chi(-q^3) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, 0, ...].
G.f.: Product_{k>0} (1 - x^(2*k-1))^3 / (1 - x^(6*k-3)).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - u * (2 - u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 2*u + 4) - v^3 * (u^2 + u + 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 * (u6^2 - u2 * u3) - u6 * (u3^2 - u6*u2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A092848.
a(n) = -3 * A124243(n) unless n=0. a(n) = (-1)^n * A132972(n).
a(2*n) = A128128(n). a(2*n + 1) = - 3* A132302(n).
Convolution inverse of A128128.
Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = (-2+2*3^(1/2))^(1/3). - Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A294387 Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 6, -12, 21, -36, 60, -96, 150, -228, 342, -504, 732, -1050, 1488, -2088, 2901, -3996, 5460, -7404, 9972, -13344, 17748, -23472, 30876, -40413, 52644, -68268, 88152, -113364, 145224, -185352, 235734, -298800, 377514, -475488, 597108, -747690, 933672

Offset: 0

Michael Somos, Oct 29 2017

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 3*x + 6*x^2 - 12*x^3 + 21*x^4 - 36*x^5 + 60*x^6 - 96*x^7 + ...

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. A128111, A128128, A132972, A164270, A164271.

a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q]^3 / QPochhammer[ q^3, -q^3], {q, 0, n}];

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

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3*eta(q^4)^3*eta(q^6)^2/(eta(q^2)^6*eta(q^3)*eta(q^12)))} \\ Altug Alkan, Mar 21 2018

Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6)^2 / (eta(q^2)^6 * eta(q^3) * eta(q^12)) in powers of q.
Expansion of (c(q) - c(q^4)) * (c(q) - 4*c(q^4)) / (c(q) + 2*c(q^4))^2 in powers of q where c(q) is a cubic AGM theta function.
Expansion of b(q^2) / b(-q) = b(q^2) / (2*b(q^4) - b(q)) in powers of q where b() is a cubic AGM theta function.
Expansion of (3*a(q^12) - a(q^4)) / (a(q) + a(q^2)) = -1/2 + 3/2*(a(-q^3) + 2*a(q^3)) / (2*a(q) + a(-q)) in powers of q where a() is a cubic AGM theta function.
Euler transform of period 12 sequence [-3, 3, -2, 0, -3, 2, -3, 0, -2, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128111.
G.f. A(q) = (1 - T(q)) / (1 + 2*T(q))  where T(q) = q*A128111(q^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v) + 3*(u*v)^2 - 4*(u*v)^3 + 2*(u*v)^4 - (u^3 + v^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u*(1 + u + u^2) - v^3*(1 - 2*u + 4*u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 + u2 + u1*u2 - u3*u6 - 2*u1*u2*u3*u6.
G.f.: Product_{k>0} (1 + x^(6*k-3)) / (1 + x^(2*k-1))^3.
a(n) = (-1)^n * A128128(n). Convolution inverse of A132972.
a(3*n + 1) = -3 * A164270(n). a(3*n + 2) = 6 * A164271(n).
Empirical : Sum_{n>=0} a(n)/exp(Pi*n) = 1/2*(2+2*3^(1/2))^(1/3), validated up to 1000 digits. - Simon Plouffe, May 06 2023

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A062244 McKay-Thompson series of class 36B for the Monster group.

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A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

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A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

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A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM theta function.

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A294387 Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

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