A022597 Expansion of Product_{m >= 1} (1 + q^m)^(-2).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• nice
• easy

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.

Examples

			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 + ...
		

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 [math.CO], 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.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A022567, A073252, A081362.

Cf. A089814 (expansion of Product_{k>=1}(1-q^(10k-5))^2).

Column k=2 of A286352.

Programs

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 *)

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 */

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