A112150 McKay-Thompson series of class 16a for the Monster group.

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

Offset: 0

Author

Michael Somos, Aug 28 2005

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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).
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A022601, A107635, A112142.

Programs

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 *)

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 */

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