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 A022597 Expansion of Product_{m >= 1} (1 + q^m)^(-2). 11
 1, -2, 1, -2, 4, -4, 5, -6, 9, -12, 13, -16, 21, -26, 29, -36, 46, -54, 62, -74, 90, -106, 122, -142, 171, -200, 227, -264, 311, -358, 408, -470, 545, -626, 709, -810, 933, -1062, 1198, -1362, 1555, -1760, 1980, -2238, 2536, -2858, 3205, -3602, 4063, -4560, 5092, -5704, 6400, -7150, 7966 (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). McKay-Thompson series of class 24J for the Monster group. REFERENCES T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe) D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449, 2018. Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13. Eric Weisstein's World of Mathematics, Jacobi Theta Functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(1/12) * (eta(q) / eta(q^2))^2 in powers of q. Euler transform of period 2 sequence [ -2, 0, ...]. - Michael Somos, Sep 10 2005 Expansion of chi(-x)^2 in powers of x where chi() is a Ramanujan theta function. G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022567. G.f.: Product_{k>0} (1 + x^k)^-2. Convolution square of A081362. Convolution inverse of A022567. a(n) = (-1)^n * A073252(n). a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015 a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017 G.f.: exp(-2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018 EXAMPLE G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 + ... T24J = 1/q - 2*q^11 + q^23 - 2*q^35 + 4*q^47 - 4*q^59 + 5*q^71 - 6*q^83 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *) a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}]^-2, {x, 0, n}]; (* Michael Somos, Jul 11 2011 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^2, n))}; /* Michael Somos, Sep 10 2005 */ CROSSREFS Cf. A022567, A073252, A081362. Cf. A089814 (expansion of Product_{k>=1}(1-q^(10k-5))^2). Column k=2 of A286352. Sequence in context: A108802 A023673 A132965 * A073252 A134005 A132320 Adjacent sequences:  A022594 A022595 A022596 * A022598 A022599 A022600 KEYWORD sign,nice,easy AUTHOR STATUS approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)