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A112212 McKay-Thompson series of class 84C for the Monster group.

%I A112212 #27 Feb 16 2025 08:32:59
%S A112212 1,1,0,1,1,1,1,2,3,2,3,3,4,4,4,6,7,7,7,9,10,12,13,14,17,18,19,22,26,
%T A112212 28,29,34,38,41,44,50,57,60,65,72,81,86,94,105,114,124,133,146,161,
%U A112212 174,187,204,224,240,258,282,309,332,354,386,419,450,481,524,569,606
%N A112212 McKay-Thompson series of class 84C for the Monster group.
%C A112212 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A112212 Given g.f. A(x), the first term of the left side of Cayley's identity is A(q). - _Michael Somos_, Dec 03 2013
%D A112212 A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.
%H A112212 G. C. Greubel, <a href="/A112212/b112212.txt">Table of n, a(n) for n = 0..1000</a>
%H A112212 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 A112212 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A112212 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A112212 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A112212 Expansion of q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - _Michael Somos_, Dec 03 2013
%F A112212 Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - _Michael Somos_, Dec 03 2013
%F A112212 G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Dec 03 2013
%F A112212 G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - _Michael Somos_, Dec 03 2013
%F A112212 a(n) = (-1)^n * A102314(n). a(2*n + 1) = A093950(n). - _Michael Somos_, Dec 03 2013
%F A112212 a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015
%e A112212 G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 3*x^10 + ...
%e A112212 T84C = 1/q + q^2 + q^8 + q^11 + q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...
%t A112212 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^7, x^14], {x, 0, n}]; (* _Michael Somos_, Dec 03 2013 *)
%t A112212 a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* _Michael Somos_, Dec 03 2013 *)
%o A112212 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* _Michael Somos_, Dec 03 2013 */
%Y A112212 Cf. A093950, A102314.
%K A112212 nonn
%O A112212 0,8
%A A112212 _Michael Somos_, Aug 28 2005