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A112150 McKay-Thompson series of class 16a for the Monster group.

%I A112150 #21 Feb 16 2025 08:32:58
%S A112150 1,6,15,26,51,102,172,276,453,728,1128,1698,2539,3780,5505,7882,11238,
%T A112150 15918,22259,30810,42438,58110,78909,106392,142770,190698,253179,
%U A112150 334266,439581,575784,750613,974316,1260336,1624702,2086530,2670162,3406695,4333590
%N A112150 McKay-Thompson series of class 16a for the Monster group.
%C A112150 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A112150 G. C. Greubel, <a href="/A112150/b112150.txt">Table of n, a(n) for n = 0..1000</a>
%H A112150 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%H A112150 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%H A112150 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A112150 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A112150 Expansion of chi(x)^6 in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Jul 03 2014
%F A112150 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 03 2014
%F A112150 Expansion of q^(1/4) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 in powers of q. - _Michael Somos_, Jul 03 2014
%F A112150 Euler transform of period 4 sequence [ 6, -6, 6, 0, ...]. - _Michael Somos_, Jul 03 2014
%F A112150 Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - u) * (u^3 - v) - 9*u*v * (-7 + 2*u*v). -  _Michael Somos_, Jul 03 2014
%F A112150 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) =  f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jul 03 2014
%F A112150 G.f.: Product_{k>0} (1 + (-x)^k)^-6 = Product_{k>0} (1 + x^(2*k - 1))^6. - _Michael Somos_, Jul 03 2014
%F A112150 Convolution square is A112142. Convolution square of A107635. - _Michael Somos_, Jul 03 2014
%F A112150 a(n) = (-1)^n * A022601(n). - _Michael Somos_, Jul 03 2014
%F A112150 a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A112150 G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018
%e A112150 G.f. = 1 + 6*x + 15*x^2 + 26*x^3 + 51*x^4 + 102*x^5 + 172*x^6 + 276*x^7 + ...
%e A112150 T16a = 1/q + 6*q^3 + 15*q^7 + 26*q^11 + 51*q^15 + 102*q^19 + 172*x^23 + ...
%t A112150 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^6, {x, 0, n}]; (* _Michael Somos_, Jul 03 2014 *)
%t A112150 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^6, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%o A112150 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6, n))}; /* _Michael Somos_, Jul 03 2014 */
%Y A112150 Cf. A022601, A107635, A112142.
%K A112150 nonn
%O A112150 0,2
%A A112150 _Michael Somos_, Aug 28 2005