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A112216 McKay-Thompson series of class 92A for the Monster group. 3

%I

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

%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.

%C Also McKay-Thompson series of class 92B for Monster. - _Michel Marcus_, Feb 19 2014

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

%H G. C. Greubel, <a href="/A112216/b112216.txt">Table of n, a(n) for n = -1..1000</a>

%H 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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%H M. 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 + 1/q * chi(q) * chi(q^23) in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Aug 11 2015

%F Expansion of -1 + (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^4) * eta(q^23) * eta(q^92)) in powers of q. - _Michael Somos_, Aug 11 2015

%F G.f.: -1 + 1/x * Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(46*k - 23)). - _Michael Somos_, Aug 11 2015

%F a(n) = A225956(n) unless n=0. a(2*n) = A092833(n). - _Michael Somos_, Aug 11 2015

%F a(n) ~ exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015

%e T92A = 1/q + q^2 + q^3 + q^4 + q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + ...

%t a[ n_] := SeriesCoefficient[ -1 + 1/q QPochhammer[ -q, q^2] QPochhammer[ -q^23, q^46], {q, 0, n}]; (* _Michael Somos_, Aug 11 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -x + (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))}; /* _Michael Somos_, Aug 11 2015 */

%Y Cf. A092833, A225956.

%K nonn,changed

%O -1,9

%A _Michael Somos_, Aug 28 2005

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Last modified November 14 07:19 EST 2019. Contains 329111 sequences. (Running on oeis4.)