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

%I A007259 M4504 #51 Feb 16 2025 08:32:31
%S A007259 1,-8,28,-64,134,-288,568,-1024,1809,-3152,5316,-8704,13990,-22208,
%T A007259 34696,-53248,80724,-121240,180068,-264448,384940,-556064,796760,
%U A007259 -1132544,1598789,-2243056,3127360,-4333568,5971922,-8188096,11170160,-15163392,20491033,-27572936
%N A007259 Expansion of Product_{m>=1} (1 + q^m)^(-8).
%C A007259 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A007259 McKay-Thompson series of class 6F for the Monster group.
%D A007259 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 118, Problem 24.
%D A007259 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007259 Seiichi Manyama, <a href="/A007259/b007259.txt">Table of n, a(n) for n = 0..10000</a>
%H A007259 J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%H A007259 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A007259 J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.
%H A007259 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A007259 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A007259 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A007259 Expansion of chi(-q)^8 in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Aug 18 2007
%F A007259 Expansion of q^(-1/3) * (eta(q) / eta(q^2))^8 in powers of q. - _Michael Somos_, Aug 18 2007
%F A007259 Euler transform of period 2 sequence [ -8, 0, ...]. - _Michael Somos_, Aug 18 2007
%F A007259 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
%F A007259 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
%F A007259 G.f.: Product_{k>0} (1 + x^k)^(-8).
%F A007259 a(2*n) = A014705(n). a(2*n + 1) = -8 * A022573(n). a(n) = A007263(3*n - 1).
%F A007259 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A007259 a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 05 2017
%F A007259 G.f.: exp(-8*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e A007259 1 - 8*x + 28*x^2 - 64*x^3 + 134*x^4 - 288*x^5 + 568*x^6 - 1024*x^7 + ...
%e A007259 T6F = 1/q - 8q^2 + 28q^5 - 64q^8 + 134q^11 - 288q^14 + 568q^17 + ...
%t A007259 a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^8, {q, 0, n}] (* _Michael Somos_, Jul 11 2011 *)
%t A007259 a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, 1, n, 2}]^8, {q, 0, n}] (* _Michael Somos_, Jul 11 2011 *)
%t A007259 a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^-8, {q, 0, n}] (* _Michael Somos_, Jul 11 2011 *)
%o A007259 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^8, n))}
%Y A007259 Cf. A007263, A014705, A022573.
%Y A007259 Column k=8 of A286352.
%K A007259 sign,easy,nice
%O A007259 0,2
%A A007259 _N. J. A. Sloane_