A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q.

1, 2, 2, 4, 5, 6, 10, 12, 15, 20, 26, 32, 40, 50, 60, 76, 92, 110, 134, 160, 191, 230, 272, 320, 380, 446, 522, 612, 715, 830, 966, 1120, 1292, 1494, 1720, 1976, 2272, 2602, 2974, 3400, 3876, 4412, 5020, 5700, 6460, 7322, 8282, 9352, 10559, 11900, 13396

Offset

0,2

Comments

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

In the notation of Dragonette on page 498 Lemma 6, the generating function is G_2(q^(1/2))/2.

Equals A000009 convolved with A010054. [Gary W. Adamson, Mar 16 2010]

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.

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. 16.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of psi(x) / chi(-x) = f(-x^2) / chi(-x)^2 = f(-x) / chi(-x)^3 = phi(-x) / chi(-x)^4 = phi(x) / chi(-x^2)^2 = f(-x^2)^2 / phi(-x) = f(-x)^4 / phi(-x)^3 = psi(x)^2 / f(-x^2) = chi(x)^2 * psi(x^2) = f(-x^2)^3 / f(-x)^2 in powers of x where f(), phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Feb 18 2006

Euler transform of period 2 sequence [ 2, -1, ...].

G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1))^2 = Product_{k>0} (1 - x^k) * (1 + x^k)^3.

a(n) = b(n)+b(n-1)+b(n-3)+b(n-6)+...+b(n-k*(k+1)/2)+..., where b() is A000009(). E.g. a(8) = b(8)+b(7)+b(5)+b(2) = 6+5+3+1 = 15. - Vladeta Jovovic, Aug 18 2004

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (3/4)^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132970. - Michael Somos, Jul 11 2015

a(n) = A053254(2*n). - Michael Somos, Jul 11 2015

a(n) ~ exp(Pi*sqrt(n/3))/(4*sqrt(n)). - Vaclav Kotesovec, Sep 07 2015

Example

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 15*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 10*q^37 + 12*q^43 + 15*q^49 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k) * (1 + x^k)^3, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 2, n, 2}] / Product[ 1 - x^k, {k, 1, n, 2}]^2, {x, 0, n}];
a[ n_] := With[ {t = Log[q]/(2 Pi I)}, SeriesCoefficient[ q^(-1/6) DedekindEta[ 2 t]^3 / DedekindEta[ t]^2, {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Jul 11 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(j j + j) / Product[ 1 + x^k, {k, 1, 2 j + 1, 2}], {j, 0, Sqrt[8 n + 1]/2}], {x, 0, 2 n}]]; (* Michael Somos, Jul 11 2015 *)

Prog

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

Crossrefs

Cf. A000009, A010054. [Gary W. Adamson, Mar 16 2010]

Cf. A053254, A132970.

Sequence in context: A211860 A250114 A056219 * A232166 A325555 A138883

Adjacent sequences: A085137 A085138 A085139 * A085141 A085142 A085143

Keyword

nonn

Author

Michael Somos, Jun 20 2003

Status

approved