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A007259
Expansion of Product_{m>=1} (1 + q^m)^(-8).
(Formerly M4504)
7
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
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
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.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
Column k=8 of A286352.
Sequence in context: A340964 A353325 A101127 * A134747 A083013 A350144
KEYWORD
sign,easy,nice
STATUS
approved