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A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).
(Formerly M4504)
5
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, -27572936 (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).

McKay-Thompson series of class 6F for the Monster group.

REFERENCES

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 118, Problem 24.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

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

Expansion of chi(-q)^8 in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Aug 18 2007

Expansion of q^(-1/3) * (eta(q) / eta(q^2))^8 in powers of q. - Michael Somos, Aug 18 2007

Euler transform of period 2 sequence [ -8, 0, ...]. - Michael Somos, Aug 18 2007

Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v^2 - u^2 * v - 16 * u. - Michael Somos, Aug 18 2007

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16 / f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 18 2007

G.f.: Product_{k>0} (1 + x^k)^(-8).

a(2*n) = A014705(n). a(2*n + 1) = -8 * A022573(n). a(n) = A007263(3*n - 1).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017

G.f.: exp(-8*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

EXAMPLE

1 - 8*x + 28*x^2 - 64*x^3 + 134*x^4 - 288*x^5 + 568*x^6 - 1024*x^7 + ...

T6F = 1/q - 8q^2 + 28q^5 - 64q^8 + 134q^11 - 288q^14 + 568q^17 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n, 2}]^8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^-8, {q, 0, n}] (* Michael Somos, Jul 11 2011 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8, n))}

CROSSREFS

Cf. A007263, A014705, A022573.

Column k=8 of A286352.

Sequence in context: A007331 A002408 A101127 * A134747 A083013 A028553

Adjacent sequences:  A007256 A007257 A007258 * A007260 A007261 A007262

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)