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A058565 McKay-Thompson series of class 21C for the Monster group. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

M. 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

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

EXAMPLE

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 + ...

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 *)

PROG

(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 */

CROSSREFS

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

Cf. A002652, A002655, A282877.

Sequence in context: A022121 A171672 A070073 * A170901 A201882 A050391

Adjacent sequences:  A058562 A058563 A058564 * A058566 A058567 A058568

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 27 2000

EXTENSIONS

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

STATUS

approved

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)