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 A132040 McKay-Thompson series of class 10B for the Monster group with a(0) = -4. 6
 1, -4, 6, -8, 17, -32, 54, -80, 116, -192, 290, -408, 585, -832, 1192, -1648, 2237, -3072, 4156, -5576, 7414, -9824, 12964, -16896, 22002, -28544, 36794, -47184, 60185, -76736, 97388, -122864, 154615, -194048, 242904, -302800, 376271, -466720, 577176, -711840 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS Seiichi Manyama, Table of n, a(n) for n = -1..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^-1 * (chi(-q) * chi(-q^5))^4 in powers of q where chi() is a Ramanujan theta function. Expansion of (eta(q) * eta(q^5) / (eta(q^2) * eta(q^10)))^4 in powers of q. Euler transform of period 10 sequence [ -4, 0, -4, 0, -8, 0, -4, 0, -4, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v*(u^2 - v) + 8*u * (v + 2). G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 16 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093861. G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(5*k)))^-4. a(n) = A058098(n) unless n = 0. a(n) = -(-1)^n * A112158(n) unless n = 0. Convolution inverse is A093831. - Michael Somos, Apr 26 2015 a(n) = -(-1)^n * A210459(n). - Michael Somos, Nov 01 2015, corrected by Vaclav Kotesovec, Sep 08 2017 a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017 EXAMPLE G.f. = 1/q - 4 + 6*q - 8*q^2 + 17*q^3 - 32*q^4 + 54*q^5 - 80*q^6 + 116*q^7 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5])^-4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *) PROG (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)))^4, n))}; CROSSREFS Cf. A058098, A093831, A112158, A210459. Sequence in context: A185292 A022599 A112160 * A210459 A346868 A279896 Adjacent sequences: A132037 A132038 A132039 * A132041 A132042 A132043 KEYWORD sign AUTHOR Michael Somos, Aug 07 2007 STATUS approved

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Last modified October 3 16:08 EDT 2023. Contains 365868 sequences. (Running on oeis4.)