A022598 Expansion of Product_{m>=1} (1+q^m)^(-3).

1, -3, 3, -4, 9, -12, 15, -21, 30, -43, 54, -69, 94, -123, 153, -193, 252, -318, 391, -486, 609, -754, 918, -1119, 1376, -1680, 2019, -2432, 2946, -3540, 4220, -5034, 6015, -7157, 8463, -9999, 11835, -13956, 16374, -19206, 22542, -26376, 30750, -35829, 41745

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 3*x + 3*x^2 - 4*x^3 + 9*x^4 - 12*x^5 + 15*x^6 - 21*x^7 + 30*x^8 + ...
G.f. = 1/q - 3*q^7 + 3*q^15 - 4*q^23 + 9*q^31 - 12*q^39 + 15*q^47 - 21*q^55 + ...
		

References

• T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^3.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• 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, Ramanujan Theta Functions

Crossrefs

Cf. A022568, A081362.

Column k=3 of A286352.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^3, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3, n))};

Formula

Expansion of chi(-x)^3 = phi(-x) / psi(x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Aug 09 2015

Expansion of q^(1/8) * (eta(q) / eta(q^2))^3 in powers of q. - Michael Somos, Apr 24 2015

Euler transform of period 2 sequence [ -3, 0, ...]. - Michael Somos, Aug 09 2015

Convolution cube of A081362. - Michael Somos, Apr 24 2015

Convolution inverse of A022568. - Michael Somos, Aug 09 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015

a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

G.f.: exp(-3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018