A014705 -id:A014705 - cp's OEIS frontend

A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).

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

N. J. A. Sloane

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.

			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 + ...

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).

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.
Michael 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

Cf. A007263, A014705, A022573.

Column k=8 of A286352.

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 *)

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

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

A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

Original entry on oeis.org

1, 6, 17, 38, 84, 172, 325, 594, 1049, 1796, 3005, 4912, 7877, 12430, 19309, 29580, 44766, 66978, 99150, 145374, 211242, 304382, 435194, 617674, 870651, 1219352, 1697283, 2348888, 3232919, 4426546, 6030872, 8177986, 11039633, 14838518, 19862613, 26482878, 35175989, 46552818, 61393694

Offset: 0

Peter Bala, Sep 29 2023

Compare with A224916 with g.f. ( Chi(sqrt(x))^4 - Chi(-sqrt(x))^4 )/(8*sqrt(x)),
A069910 with g.f. ( Chi(sqrt(x)) + Chi(-sqrt(x)) )/2,
A069911 with g.f. ( Chi(sqrt(x)) - Chi(-sqrt(x)) )/2,
A226622 with g.f. ( Chi(sqrt(x))^2 + Chi(-sqrt(x))^2 )/2 and
A226635 with g.f. ( Chi(sqrt(x))^2 - Chi(-sqrt(x))^2 )/(4*sqrt(x)),
Jacobi's "aequatio identica satis abstrusa" is the identity ( Chi(sqrt(x))^8 - Chi(-sqrt(x))^8 )/(16*sqrt(x)) =  Product_{k >= 1} (1 + x^k)^8.

Cf. A000700, A014705, A069910, A069911, A224916, A226622, A226635.

with(QDifferenceEquations):
 seq(coeff((1/2)*expand(QPochhammer(-q,q^2,40)^4 + QPochhammer(q,q^2,40)^4), q, 2*n), n = 0..40);
#alternative program
seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);

nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^2 * (1 + x^(2*k-1))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 29 2025 *)

G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.
G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025

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A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).

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