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A102314 McKay-Thompson series of class 42C for the Monster group.

%I A102314 #32 Feb 16 2025 08:32:55
%S A102314 1,-1,0,-1,1,-1,1,-2,3,-2,3,-3,4,-4,4,-6,7,-7,7,-9,10,-12,13,-14,17,
%T A102314 -18,19,-22,26,-28,29,-34,38,-41,44,-50,57,-60,65,-72,81,-86,94,-105,
%U A102314 114,-124,133,-146,161,-174,187,-204,224,-240,258,-282,309,-332,354,-386,419,-450,481,-524,569,-606,651,-703
%N A102314 McKay-Thompson series of class 42C for the Monster group.
%C A102314 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A102314 Given g.f. A(x), the second term of the left side of Cayley's identity is -A(q). - _Michael Somos_, Dec 03 2013
%D A102314 A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
%H A102314 G. C. Greubel, <a href="/A102314/b102314.txt">Table of n, a(n) for n = 0..1000</a>
%H A102314 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 A102314 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A102314 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A102314 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A102314 Expansion of chi(-x) * chi(-x^7) in powers of x where chi() is a Ramanujan theta function.
%F A102314 Expansion of q^(1/3) * eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)) in powers of q.
%F A102314 Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, ...].
%F A102314 Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^2*v - 2*u.
%F A102314 G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093950.
%F A102314 G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^(7*k))).
%F A102314 a(n) = (-1)^n * A112212(n). a(2*n + 1) = - A093950(n). a(4*n) = A193826(n). a(4*n + 2) = A193883(n).
%F A102314 Convolution inverse is A093950.
%F A102314 a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017
%e A102314 G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - 2*x^7 + 3*x^8 - 2*x^9 + 3*x^10 - 3*x^11 + ...
%e A102314 T42C = 1/q - q^2 - q^8 + q^11 - q^14 + q^17 - 2*q^20 + 3*q^23 - 2*q^26 + ...
%t A102314 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, n}]; (* _Michael Somos_, Aug 06 2011 *)
%t A102314 a[ n_] := SeriesCoefficient[ 1 / ( Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}] ), {x, 0, n}]; (* _Michael Somos_, Aug 06 2011 *)
%o A102314 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)), n))};
%Y A102314 Cf. A093950, A112212, A193826, A193883.
%K A102314 sign
%O A102314 0,8
%A A102314 _Michael Somos_, Jan 03 2005