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A101127 McKay-Thompson series of class 12D for the Monster group. 3
1, 8, 28, 64, 134, 288, 568, 1024, 1809, 3152, 5316, 8704, 13990, 22208, 34696, 53248, 80724, 121240, 180068, 264448, 384940, 556064, 796760, 1132544, 1598789, 2243056, 3127360, 4333568, 5971922, 8188096, 11170160, 15163392, 20491033 (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).
REFERENCES
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Infinite Product
FORMULA
Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [8, -8, 8, 0, ...].
Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16.
G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8.
A007259(n) = (-1)^n * a(n). Convolution square of A112160.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 12 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018
EXAMPLE
T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* Michael Somos, Sep 12 2017 *)
PROG
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))};
(PARI) {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))};
CROSSREFS
Sequence in context: A002408 A340964 A353325 * A007259 A134747 A083013
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 02 2004
STATUS
approved

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Last modified June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)