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A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

%I A022601 #36 Feb 16 2025 08:32:34
%S A022601 1,-6,15,-26,51,-102,172,-276,453,-728,1128,-1698,2539,-3780,5505,
%T A022601 -7882,11238,-15918,22259,-30810,42438,-58110,78909,-106392,142770,
%U A022601 -190698,253179,-334266,439581,-575784,750613,-974316,1260336,-1624702,2086530,-2670162
%N A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).
%C A022601 McKay-Thompson series of class 8F for the Monster group.
%C A022601 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A022601 Seiichi Manyama, <a href="/A022601/b022601.txt">Table of n, a(n) for n = 0..1000</a>
%H A022601 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 A022601 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 A022601 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 A022601 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A022601 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A022601 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A022601 Expansion of chi(-x)^6 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Jul 01 2014
%F A022601 Expansion of q^(1/4) * 2 * k'(q) / k(q)^(1/2) in powers of q where k() is the elliptic modulus. - _Michael Somos_, Jul 01 2014
%F A022601 Expansion of q^(1/4) * (eta(q) / eta(q^2))^6 in powers of q. - _Michael Somos_, Jul 01 2014
%F A022601 Euler transform of period 2 sequence [ -6, 0, ...]. - _Michael Somos_, Jul 01 2014
%F A022601 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (21 + 6*u*v). - _Michael Somos_, Jul 01 2014
%F A022601 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022571. - _Michael Somos_, Jul 01 2014
%F A022601 Convolution inverse of A022571. Convolution sixth power of A081362. - _Michael Somos_, Jul 01 2014
%F A022601 a(n) = (-1)^n * A112150(n) = A058088(2*n) = A112145(2*n). - _Michael Somos_, Jul 01 2014
%F A022601 a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A022601 a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 05 2017
%F A022601 G.f.: exp(-6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e A022601 G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 51*x^4 - 102*x^5 + 172*x^6 - 276*x^7 + ...
%e A022601 T8F = 1/q - 6*q^3 + 15*q^7 - 26*q^11 + 51*q^15 - 102*q^19 + 172*q^23 + ...
%t A022601 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^6, {x, 0, n}]; (* _Michael Somos_, Jul 01 2014 *)
%t A022601 nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%o A022601 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^6, n))}; /* _Michael Somos_, Jul 01 2014 */
%Y A022601 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A022601 Cf. A022571, A058088, A081362, A112145, A112150.
%Y A022601 Column k=6 of A286352.
%K A022601 sign
%O A022601 0,2
%A A022601 _N. J. A. Sloane_