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A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)*(1+x^(3k))*(1+x^(5k))). 2
1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 768, 1040, 1397, 1864, 2470, 3254, 4261, 5550, 7192, 9277, 11911, 15229, 19391, 24597, 31085, 39150, 49142, 61489, 76702, 95401, 118324, 146362, 180573, 222226, 272826, 334173, 408394, 498022 (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).

LINKS

Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2001 terms from Vaclav Kotesovec)

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. 17.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

a(n) ~ exp(Pi*sqrt(37*n/5)/3) * sqrt(37) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015

G.f.: (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015

Euler transform of period 30 sequence [ 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, ...]. - Michael Somos, Mar 07 2016

Expansion of chi(-x^3) * chi(-x^5) / phi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Mar 07 2016

a(n) - A035939(2*n + 1) = A122129(2*n + 1). - Michael Somos, Mar 07 2016

EXAMPLE

G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 46*x^7 + ...

G.f. = q^-1 + 2*q^2 + 4*q^5 + 7*q^8 + 12*q^11 + 19*q^14 + 30*q^17 + 46*q^20 + ...

MAPLE

series(product((1+x^k)/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), k=1..100), x=0, 100);

MATHEMATICA

CoefficientList[ Series[ Product[(1 + x^k)/((1 - x^k)*(1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 45}], x] (* Robert G. Wilson v, Jan 12 2005 *)

nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-1))*(1+x^(5*k-2))*(1+x^(5*k-3))*(1+x^(5*k-4)) / ((1-x^(6*k))*(1-x^(3*k-1))*(1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 07 2016 *)

PROG

(PARI) q='q+O('q^33); E(k)=eta(q^k);

Vec( (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

CROSSREFS

Cf. A035939, A098151, A102346, A122129.

Sequence in context: A000070 A008609 A264392 * A102346 A333148 A326080

Adjacent sequences:  A100820 A100821 A100822 * A100824 A100825 A100826

KEYWORD

nonn

AUTHOR

Noureddine Chair, Jan 06 2005

EXTENSIONS

More terms from Robert G. Wilson v, Jan 12 2005

Offset corrected by Vaclav Kotesovec, Sep 01 2015

a(14) = 563 <- 562 corrected by Vaclav Kotesovec, Sep 01 2015

STATUS

approved

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Last modified July 2 09:21 EDT 2020. Contains 335398 sequences. (Running on oeis4.)