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A112150 McKay-Thompson series of class 16a for the Monster group. 2
1, 6, 15, 26, 51, 102, 172, 276, 453, 728, 1128, 1698, 2539, 3780, 5505, 7882, 11238, 15918, 22259, 30810, 42438, 58110, 78909, 106392, 142770, 190698, 253179, 334266, 439581, 575784, 750613, 974316, 1260336, 1624702, 2086530, 2670162, 3406695, 4333590 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for McKay-Thompson series for Monster simple group

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

FORMULA

Expansion of chi(x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 03 2014

Expansion of q^(1/4) * 2 * (k(q) * k'(q))^(-1/2) in powers of q where k() is the elliptic modulus. - Michael Somos, Jul 03 2014

Expansion of q^(1/4) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 in powers of q. - Michael Somos, Jul 03 2014

Euler transform of period 4 sequence [ 6, -6, 6, 0, ...]. - Michael Somos, Jul 03 2014

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - u) * (u^3 - v) - 9*u*v * (-7 + 2*u*v). -  Michael Somos, Jul 03 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) =  f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 03 2014

G.f.: Product_{k>0} (1 + (-x)^k)^-6 = Product_{k>0} (1 + x^(2*k - 1))^6. - Michael Somos, Jul 03 2014

Convolution square is A112142. Convolution square of A107635. - Michael Somos, Jul 03 2014

a(n) = (-1)^n * A022601(n). - Michael Somos, Jul 03 2014

a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

EXAMPLE

G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 51*x^4 + 102*x^5 + 172*x^6 + 276*x^7 + ...

T16a = 1/q + 6*q^3 + 15*q^7 + 26*q^11 + 51*q^15 + 102*q^19 + 172*x^23 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 03 2014 *)

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^6, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6, n))}; /* Michael Somos, Jul 03 2014 */

CROSSREFS

Cf. A022601, A107635, A112142.

Sequence in context: A008440 A284629 A022601 * A240948 A072257 A227952

Adjacent sequences:  A112147 A112148 A112149 * A112151 A112152 A112153

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 28 2005

STATUS

approved

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Last modified January 21 12:00 EST 2019. Contains 319354 sequences. (Running on oeis4.)