A138688 - OEIS

%I #20 Feb 16 2025 08:33:08

%S 1,2,4,6,11,18,28,42,62,90,128,180,250,342,464,624,831,1098,1440,1878,

%T 2432,3132,4012,5112,6485,8190,10300,12900,16097,20016,24804,30636,

%U 37724,46314,56700,69228,84302,102402,124088,150024,180973,217836,261664,313680

%N McKay-Thompson series of class 24I for the Monster group with a(0) = 2.

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

%C Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

%H G. C. Greubel, <a href="/A138688/b138688.txt">Table of n, a(n) for n = -1..1000</a>

%H K. Bringmann and H. Swisher, <a href="http://dx.doi.org/10.1090/S0002-9939-07-08735-7">On a conjecture of Koike on identities between Thompson series and Roger-Ramanujan functions</a>, Proc. Amer. Math. Soc. 135 (2007), 2317-2326. See page 2325 (A.7).

%H J. H. Conway and S. P. Norton, <a href="https://doi.org/10.1112/blms/11.3.308">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015

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

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

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

%F Expansion of psi(q^4) * phi(-q^3) / (phi(-q) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.

%F Expansion of eta(q^2) * eta(q^3)^2 * eta(q^8)^2 * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6) * eta(q^24)^2) in powers of q.

%F Euler transform of period 24 sequence [ 2, 1, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 2, 2, 0, 1, 2, 0, ...].

%F G.f.: (G(x) * G(x^24) + x^5 * H(x) * H(x^24))^2 * (G(x^4) * G(x^6) + x^2 * H(x^4) * H(x^6)) where G() and H() are Rogers-Ramanujan functions.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = A058579(n) unless n=0.

%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015

%e G.f. = 1/q + 2 + 4*q + 6*q^2 + 11*q^3 + 18*q^4 + 28*q^5 + 42*q^6 + 62*q^7 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^2] EllipticTheta[ 4, 0, q^3] / (EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^6]), {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)

%t nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(3*k))^2 * (1-x^(4*k)) * (1+x^(4*k))^2 * (1+x^(6*k)) / ((1-x^k) * (1-x^(24*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};

%Y Cf. A058579.

%K nonn,changed

%O -1,2

%A _Michael Somos_, Mar 26 2008