A058565 McKay-Thompson series of class 21C for the Monster group.

1, 3, 8, 11, 25, 35, 57, 86, 139, 198, 291, 417, 588, 812, 1132, 1538, 2103, 2805, 3767, 4963, 6554, 8548, 11165, 14426, 18601, 23830, 30443, 38642, 48986, 61748, 77669, 97206, 121478, 151067, 187556, 231974, 286385, 352340, 432641, 529688, 647241, 788738, 959470, 1164291, 1410386

Offset: 0

Author

N. J. A. Sloane, Nov 27 2000

Keywords

• nonn

Examples

			G.f. = 1 + 3*x + 8*x^2 + 11*x^3 + 25*x^4 + 35*x^5 + 57*x^6 + 86*x^7 + ... -  _Michael Somos_, Feb 26 2017
T21C = 1/q + 3*q^2 + 8*q^5 + 11*q^8 + 25*q^11 + 35*q^14 + 57*q^17 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 176 Entry 32(iii).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Cf. A002652, A002655, A282877.

Programs

Mathematica

a[ n_] := With[ {A = (QPochhammer[ x^7] / QPochhammer[ x])^4}, SeriesCoefficient[ (1/A + 13 x + 49 x^2 A)^(1/3), {x, 0, n}]]; (*  Michael Somos, Feb 26 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(1/3)*(eta[q]*eta[q^7]/(eta[q^2] *eta[q^14])); a:= CoefficientList[Series[(A + 4*q/A^2), {q,0,60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 21 2018 *)
a[ n_] := With[ {A1 = QPochhammer[ x] QPochhammer[ x^7], A2 = QPochhammer[ x^2] QPochhammer[ x^14]}, SeriesCoefficient[ (A1^3 + 4 x A2^3) / (A1^2 A2), {x, 0, n}]]; (*  Michael Somos, Oct 27 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1/A + 13*x + 49*x^2 * A)^(1/3), n))}; /*  Michael Somos, Feb 26 2017 */

PARI

q='q+O('q^50); A = (eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))); Vec(A + 4*q/A^2) \\ G. C. Greubel, Jun 21 2018

PARI

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1^2 * A2), n))}; /* Michael Somos, Oct 27 2018 */

Formula

From Michael Somos, Feb 26 2017: (Start)

Expansion of f(-x^7, -x^14)^2 / f(-x, -x^2) * (w3/w1^2 + x*w2/w3^2 - x*w1/w2^2) in powers of x where w1 = f(-x, -x^6), w2 = f(-x^2, -x^5), w3 = f(-x^3, -x^4) and f(, ) is Ramanujan's general theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = f(t) where q = exp(2 Pi i t).

Convolution cube is A282877.

Convolution product with A002655 is A002652. (End)

Expansion of A + 4*q/A^2, where A = q^(1/3)*(eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))), in powers of q. - G. C. Greubel, Jun 21 2018

a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Feb 26 2017

Extensions

Terms a(8) onward added by G. C. Greubel, Jun 21 2018