|
%I #19 Jun 28 2018 11:01:49
%S 1,1,4,7,13,19,33,47,74,106,154,214,307,417,575,772,1045,1379,1837,
%T 2394,3135,4048,5232,6686,8560,10840,13737,17273,21701,27086,33783,
%U 41890,51893,63969,78748,96536,118196,144146,175561,213122,258327,312202
%N McKay-Thompson series of class 23A for the Monster group with a(0) = 1.
%C A058570(n) = a(n) unless n=0.
%H G. C. Greubel, <a href="/A134781/b134781.txt">Table of n, a(n) for n = -1..1000</a>
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787871">Mathieu group M24 and modular forms</a>, Nagoya Math. J., 99 (1985), 147-157. MR0805086 (87e:11060)
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Associated with permutations in Mathieu group M24 of shape (23)(1).
%F G.f. is Fourier series of a level 23 modular function. f(-1/ (23 t)) = f(t) where q = exp(2 Pi i t).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u^2 - v) + 2*(u*v * (u + v) + 2*(u^2 + v^2) + 5*u*v + 3*(u + v) + 1).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (2 - u - w) + v*(9 + 2*(u + w)) + u^2 + u*w + w^2 + 4*(u + w) + 6.
%F G.f.: (Sum_{j,k} x^(2*j^2 + j*k + 3*k^2)) / (x * Product_{k>0} (1 - x^k) * (1 - x^(23*k))).
%F a(n) ~ exp(4*Pi*sqrt(n/23)) / (sqrt(2) * 23^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018
%e 1/q + 1 + 4*q + 7*q^2 + 13*q^3 + 19*q^4 + 33*q^5 + 47*q^6 + 74*q^7 + ...
%t nmax = 40; QP = QPochhammer; s = Sum[x^(2*j^2 + j*k + 3*k^2), {j, -nmax, nmax}, {k, -nmax, nmax}]/(QP[x]*QP[x^23]) + O[x]^nmax; CoefficientList[s, x] (* _Jean-François Alcover_, Nov 15 2015 *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; e46A := (eta[q]*eta[q^23]/(eta[q^2] *eta[q^46])); T23A := (e46A + 1)*(e46A^2 + 4)/e46A^2; Table[ SeriesCoefficient[1 + T23A, {q, 0, n}], {n,-1,50}] (* _G. C. Greubel_, Feb 13 2018 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (1 + 2 * x * Ser(qfrep([4, 1; 1, 6], n, 1))) / (eta(x + A) * eta(x^23 + A)), n))}
%Y Convolution with A030199 is A028930.
%K nonn
%O -1,3
%A _Michael Somos_, Nov 12 2007
|
