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 A062242 McKay-Thompson series of class 18D for the Monster group. 14
 1, 1, 1, -1, -1, 0, 1, 2, 0, -2, -3, -1, 4, 4, 1, -4, -6, -1, 5, 8, 1, -8, -10, -2, 11, 14, 4, -14, -19, -4, 17, 24, 4, -23, -31, -6, 31, 40, 9, -38, -50, -10, 46, 63, 11, -60, -79, -16, 77, 98, 21, -92, -122, -24, 112, 150, 28, -140, -183, -36, 173, 224, 46, -208, -273, -54, 249, 329, 62, -304, -396, -78, 370 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). Number 7 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014 There is a typo in the entry for this q-series in Table I of Yang 2004. The exponent of 18 should be 3. - Michael Somos, Jul 21 2014 A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(18). [Yang 2004] - Michael Somos, Jul 21 2014 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel) D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1. FORMULA G.f.: Product_{k>0} (1 - x^(6*k - 3))^3 / (1 - x^(2*k-1)). - Michael Somos, Mar 17 2004 Expansion of chi(-q^3)^3 / chi(-q) in powers of q where chi() is a Ramanujan theta function. Expansion of q^(1/3) * c(q) / c(q^2) in powers of q where c() is a cubic AGM theta function. - Michael Somos, Oct 17 2006 Expansion of q^(1/3) * eta(q^2) * eta(q^3)^3 / (eta(q) * eta(q^6)^3) in powers of q. - Michael Somos, Mar 05 2004 Euler transform of period 6 sequence [ 1, 0, -2, 0, 1, 0, ...]. - Michael Somos, Mar 05 2004 Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 2*u  +v^2 - u^2*v. - Michael Somos, Mar 17 2004 Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - v + v^2) * u^3 - (4 + 2*v + v^2) * v. - Michael Somos, Aug 11 2007 Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^5)) where f(u, v) = u^6 + v^6 - u^5*v^5 + 5*u^4*v^4 - 20*u^3*v^3 + 20*u^2*v^2 - 16*u*v + 5*u^2*v^5 + 5*u^5*v^2 - 10*u^4*v - 10*u*v^4. - Michael Somos, Aug 11 2007 G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128128. G.f.: 1 + x*(1+x)/(1 + x^2*(1+x^2)/(1 + x^3*(1+x^3)/(1 + x^4*(1+x^4)/(1 + x^5*(1+x^5)/(1 + ...))))), a continued fraction. - Paul D. Hanna, Jul 09 2013 a(n) = (-1)^n * A062244(n). a(2*n) = A132179(n). a(2*n + 1) = A092848(n). Convolution inverse of A092848. EXAMPLE G.f. = 1 + x + x^2 - x^3 - x^4 + x^6 + 2*x^7 - 2*x^9 - 3*x^10 - x^11 + 4*x^12 + ... T18D = 1/q + q^2 + q^5 - q^8 - q^11 + q^17 + 2*q^20 - 2*q^26 - 3*q^29 - x^32 +  ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3]^3 / (QPochhammer[ x] QPochhammer[ x^6]^3), {x, 0, n}]; (* Michael Somos, Jul 21 2014 *) PROG (PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=3; A = subst(A, x, x^3) / x; A = (x*A * (4 + 2*A + A^2) / (1 - A + A^2))^(1/3)); polcoeff(A, n))}; (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3), n))}; (PARI) /* Continued Fraction Expansion: */ {a(n)=local(CF); CF=1+x; for(k=0, n, CF=1 + x^(n-k+1)*(1 + x^(n-k+1))/(CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 09 2013 CROSSREFS Cf. A062244, A092848, A132179. Sequence in context: A082490 A328591 A210635 * A062244 A169979 A079957 Adjacent sequences:  A062239 A062240 A062241 * A062243 A062244 A062245 KEYWORD sign AUTHOR N. J. A. Sloane, Jun 30 2001 STATUS approved

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Last modified June 23 10:33 EDT 2021. Contains 345397 sequences. (Running on oeis4.)