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McKay-Thompson series of class 92A for the Monster group with a(0) = 1.
2

%I #25 Mar 12 2021 22:24:47

%S 1,1,0,1,1,1,1,1,2,2,2,2,3,3,3,4,5,5,5,6,7,8,8,10,12,12,13,15,17,18,

%T 19,22,25,27,28,32,36,38,41,46,51,54,58,64,71,76,81,89,99,105,112,123,

%U 134,143,153,167,182,194,207,225,244,260,277,301,325,346

%N McKay-Thompson series of class 92A for the Monster group with a(0) = 1.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A225956/b225956.txt">Table of n, a(n) for n = -1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of 1/q * chi(q) * chi(q^23) in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^4) * eta(q^23) * eta(q^92)) in powers of q.

%F Euler transform of a period 92 sequence.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - 1)^2 - u*v * (2 - 2*v + v^2 - u) * (2 - 2*u + u^2 - v) / 2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (92 t)) = f(t) where q = exp(2 Pi i t).

%F G.f.: 1/x * Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(46*k - 23)).

%F a(n) = A112216(n) unless n=0. -(-1)^n * a(n) = A132322(n).

%F a(n) ~ exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 30 2017

%e G.f. = 1/q + 1 + q^2 + q^3 + q^4 + q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + 2*q^10 + ...

%t a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q, q^2] QPochhammer[ -q^23, q^46], {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ 1/q Product[ 1 + q^k, {k, 1, n + 1, 2}] Product[ 1 + q^k, {k, 23, n + 1, 46}], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^46 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^23 + A) * eta(x^92 + A)), n))};

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, 1 + k%2 * x^k, 1 + A) * prod( k=1, n\23, 1 + k%2 * x^(23*k), 1 + A), n))};

%Y Cf. A058688, A112216, A132322.

%K nonn

%O -1,9

%A _Michael Somos_, May 21 2013