A051273 -id:A051273 - cp's OEIS frontend

A058092 McKay-Thompson series of class 9a for the Monster group.

1, 14, 65, 156, 456, 1066, 2250, 4720, 9426, 17590, 32801, 58904, 102650, 176646, 298066, 491792, 803923, 1293450, 2051156, 3221716, 5004028, 7682744, 11703580, 17663312, 26423351, 39248618, 57866503, 84685920, 123188502, 178054416, 255782770, 365467216

Offset: 0

N. J. A. Sloane, Nov 27 2000

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

In volume 2 of Raamunjuan's Notebooks is an obscure equation involving t(1-t) on the left and GG' on the right and they both are equal to the g.f. of 1/3 of this sequence. Here t^(1/3) = c(x)/a(x), (1-t)^(1/3) = b(x)/a(x) since a(x)^3 = b(x)^3 + c(x)^3. N.B. The left side was (t(1-t))^(1/3) but the exponent should be (-1/3) instead which is why the equation was so obscure. - Michael Somos, Mar 13 2019

			G.f. = 1 + 14*x + 65*x^2 + 156*x^3 + 456*x^4 + 1066*x^5 + 2250*x^6 + 4720*x^7 + ...
T9a = 1/q + 14*q^2 + 65*q^5 + 156*q^8 + 456*q^11 + 1066*q^14 + 2250*q^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
G. Manco, How to calculate moduli alpha_3n of the Ramanujan's q_3 theory, Mathematics StackExchange, Jan 2017.
Index entries for McKay-Thompson series for Monster simple group

Cf. A030197, A051273.

a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[((1 + 27*x*A)^2/A)^(1/3), n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)
CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^2 / (QPochhammer[x, x]^2*QPochhammer[x^3, x^3]^4), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/3), n))}; /* Michael Somos, Jun 16 2012 */

Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/3) in powers of x where b(), c() are cubic AGM theta functions, Michael Somos, Jun 16 2012
Convolution cube is A030197.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2015

A258941 Convolution inverse of A058537.

Original entry on oeis.org

1, -7, 41, -253, 1555, -9532, 58463, -358600, 2199546, -13491360, 82752059, -507576937, 3113328401, -19096245457, 117130782240, -718445946527, 4406737223117, -27029636742811, 165791883077354, -1016918901125280, 6237482995373629, -38258895644996020

Offset: 0

Vaclav Kotesovec, Nov 07 2015

sign

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

Convolution square is A328785. - Michael Somos, Nov 02 2019

			G.f. = 1 - 7*x + 41*x^2 - 253*x^3 + 1555*x^4 - 9532*x^5 + ... - _Michael Somos_, Nov 02 2019
G.f. = q - 7*q^7 + 41*q^13 - 253*q^19 + 1555*q^25 - 9532*q^31 + ... - _Michael Somos_, Nov 02 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..1260

Cf. A058537, A051273, A058092, A115784, A258942, A328785.

CoefficientList[Series[QPochhammer[x, x] * QPochhammer[x^3, x^3]^2 / (QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3), {x, 0, 50}], x]
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3), {q,0,60}], q] (* G. C. Greubel, Jun 22 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[x] QPochhammer[x^3]^2 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3), {x, 0, n}]; (* Michael Somos, Nov 02 2019 *)

q='q+O('q^50); A = eta(q)*eta(q^3)^2/(eta(q)^3 + 9*q*eta(q^9)^3); Vec(A) \\ G. C. Greubel, Jun 22 2018

a(n) ~ (-1)^n * c * exp(Pi*n/sqrt(3)), where c = A258942 = 8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / Gamma(1/6)^3 =  1.09786330972731096865822482325074133091288... . - Vaclav Kotesovec, Nov 14 2015
Expansion of q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3) in powers of q. - G. C. Greubel, Jun 22 2018
Expansion of q^(-1/6) * 3^(-1/2) * sqrt(b(q)*c(q))/a(q) in powers of q where a(), b(), c() are cubic AGM functions. - Michael Somos, Nov 02 2019

A258942 Decimal expansion of a constant related to A258941.

Original entry on oeis.org

1, 0, 9, 7, 8, 6, 3, 3, 0, 9, 7, 2, 7, 3, 1, 0, 9, 6, 8, 6, 5, 8, 2, 2, 4, 8, 2, 3, 2, 5, 0, 7, 4, 1, 3, 3, 0, 9, 1, 2, 8, 8, 0, 8, 7, 3, 8, 9, 3, 6, 3, 0, 4, 5, 7, 9, 1, 6, 4, 9, 8, 2, 5, 9, 9, 4, 0, 9, 2, 7, 3, 8, 5, 2, 4, 9, 4, 3, 2, 4, 8, 1, 7, 1, 8, 3, 6, 1, 6, 0, 2, 3, 7, 2, 1, 4, 3, 1, 0, 1, 7, 7, 4, 8, 1

Offset: 1

Vaclav Kotesovec, Nov 07 2015

			1.09786330972731096865822482325074133091288087389363045791649825994...

Cf. A051273, A115784, A258941.

evalf(8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / GAMMA(1/6)^3, 120); # Vaclav Kotesovec, Nov 14 2015

RealDigits[8*E^(Pi/(6*Sqrt[3]))*Pi^(5/2)/Gamma[1/6]^3, 10, 105][[1]] (* Vaclav Kotesovec, Nov 14 2015 *)

Equals limit n->infinity A258941(n) * (-1)^n / exp(Pi*n/sqrt(3)).

Equals 8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / Gamma(1/6)^3. - Vaclav Kotesovec, Nov 14 2015

A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.

Original entry on oeis.org

1, -14, 131, -1080, 8333, -61580, 441893, -3104136, 21454675, -146408362, 988876484, -6622729128, 44039891747, -291092877360, 1914072126008, -12529113820200, 81687913345362, -530724605541430, 3437326269162720, -22199991545327616, 143016156285625823

Offset: 0

Michael Somos, Nov 02 2019

sign

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

Convolution square of A258941. Convolution inverse of A058092 with more information there.

			G.f. = 1 - 14*x + 131*x^2 - 1080*x^3 + 8333*x^4 - 61580*x^5 + ...
G.f. = q - 14*q^4 + 131*q^7 - 1080*q^10 + 8333*q^13 - 61580*q^16 + ...

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998.

Cf. A058092, A258941, A051273.

a[ n_] := SeriesCoefficient[ QPochhammer[x]^2 QPochhammer[x^3]^4 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3)^2, {x, 0, n}];

{a(n) = my(A); if( n < 0, 0, A = x * O(x^n); A = ((eta(x + A) * eta(x^3 + A)) / (eta(x^2 + A) * eta(x^6 + A)))^2; polcoeff( 1 / (A + x * 16/A^2), n))};

Coefficients of 1/3 of power series in equation (13.23), page 179, [Berndt 1998]
A051273(n) = 3*a(n).
a(n) ~ (-1)^n * 64 * Pi^5 * n * exp(Pi*(3*n+1)/3^(3/2)) / Gamma(1/6)^6. - Vaclav Kotesovec, Nov 08 2019

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A058092 McKay-Thompson series of class 9a for the Monster group.

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A258941 Convolution inverse of A058537.

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A258942 Decimal expansion of a constant related to A258941.

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A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.

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