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 A132322 McKay-Thompson series of class 46A for the Monster group with a(0) = -1. 4
 1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -10, 12, -12, 13, -15, 17, -18, 19, -22, 25, -27, 28, -32, 36, -38, 41, -46, 51, -54, 58, -64, 71, -76, 81, -89, 99, -105, 112, -123, 134, -143, 153, -167, 182, -194, 207, -225, 244, -260, 277, -301, 325, -346, 369, -398, 429, -458 (list; graph; refs; listen; history; text; internal format)
 OFFSET -1,9 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 = -1..1000 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^23) in powers of q where chi() is a Ramanujan theta function. Expansion of eta(q) * eta(q^23) / (eta(q^2) * eta(q^46)) in powers of q. Euler transform of period 46 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...]. G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 2 * u + 2 * u * v. G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A092833. G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(23*k)))^-1. a(n) = A058688(n) unless n = 0. Convolution inverse of A092833. - Michael Somos, Mar 23 2015 a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018 EXAMPLE G.f. = 1/q - 1 - q^2 + q^3 - q^4 + q^5 - q^6 + 2*q^7 - 2*q^8 + 2*q^9 - 2*q^10 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^23, q^46] / q, {q, 0, n}]; (* Michael Somos, Mar 23 2015 *) PROG (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^23 + A) / (eta(x^2 + A) * eta(x^46 + A)), n))}; CROSSREFS Cf. A058688, A092833. Sequence in context: A112216 A225956 A058688 * A018118 A029084 A032229 Adjacent sequences:  A132319 A132320 A132321 * A132323 A132324 A132325 KEYWORD sign AUTHOR Michael Somos, Aug 18 2007 STATUS approved

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Last modified January 22 15:13 EST 2022. Contains 350481 sequences. (Running on oeis4.)