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 A058539 McKay-Thompson series of class 18d for the Monster group. 3
 1, 4, 10, 20, 35, 60, 100, 164, 261, 400, 600, 884, 1291, 1864, 2656, 3740, 5205, 7184, 9842, 13388, 18082, 24244, 32300, 42784, 56378, 73928, 96466, 125284, 161981, 208568, 267524, 341880, 435343, 552424, 698666, 880848, 1107229, 1387804, 1734624, 2162248 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (chi(-x^3) / chi(-x))^4 in powers of x where chi() is a Ramanujan theta function. Expansion of q^(1/3) * c(q) * b(q^2) / (b(q) * c(q^2)) in powers of q where b(), c() are cubic AGM theta functions. Expansion of q^(1/3) * (eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)))^4 in powers of q. Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 8 * (u * v)^2 - (1 + u * v) * (u^2 - v) * (v^2 - u). Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 9 * (u * v)^2 - (u - v^2 + u^2*v) * (v - u^2 + u*v^2). Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = 8 * u * v * w - (u^2 - v) * (w^2 - v). Given g.f. A(x), then B(x) = A(x^3) / x satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u*v * (1 + 25 * u*v + u^2*v^2)^2 - (u^3 + v^3 + 10 * u*v * (1 + u*v))^2. G.f. is a period 1 Fourier series which satisfies f(-1 / (54 t)) = f(t) where q = exp(2 Pi i t). Convolution square of A103262. Convolution fourth power of A003105. a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015 EXAMPLE 1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 60*x^5 + 100*x^6 + 164*x^7 + ... T18d = 1/q + 4*q^2 + 10*q^5 + 20*q^8 + 35*q^11 + 60*q^14 + 100*q^17 + ... MATHEMATICA nmax = 50; CoefficientList[Series[Product[((1+x^(3*k-1))*(1+x^(3*k-2)))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *) QP = QPochhammer; s = (QP[q^2]*(QP[q^3]/(QP[q]*QP[q^6])))^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^4, n))} /* Michael Somos, Mar 04 2012 */ CROSSREFS Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. Cf. A003105, A103262. Sequence in context: A137359 A134987 A261636 * A008112 A301153 A301008 Adjacent sequences:  A058536 A058537 A058538 * A058540 A058541 A058542 KEYWORD nonn AUTHOR N. J. A. Sloane, Nov 27 2000 STATUS approved

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Last modified October 17 22:05 EDT 2019. Contains 328134 sequences. (Running on oeis4.)