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A085140 Expansion of q^(-1/6) * eta(q^2)^3 / eta(q)^2 in powers of q. 7
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 (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).

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.

M. 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

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)