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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
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.
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
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