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A187100 Expansion of q * (psi(-q^3) * psi(q^6)) / (psi(-q) * phi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions. 6
1, 3, 7, 15, 30, 57, 104, 183, 313, 522, 852, 1365, 2150, 3336, 5106, 7719, 11538, 17067, 25004, 36306, 52280, 74700, 105960, 149277, 208951, 290706, 402127, 553224, 757158, 1031166, 1397744, 1886151, 2534316, 3391254, 4520112, 6002007, 7940846 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Johannes Bl├╝mlein, Iterative Non-iterative Integrals in Quantum Field Theory, arXiv:1808.08128 [hep-th], 2018.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^12)^3 / (eta(q)^3 * eta(q^4) * eta(q^6)^2) in powers of q.

Euler transform of period 12 sequence [ 3, 1, 2, 2, 3, 2, 3, 2, 2, 1, 3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/12) * 1/f(t) where q = exp(2 Pi i t).

Convolution inverse of A187130. A186924(n) = 4 * a(n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

EXAMPLE

q + 3*q^2 + 7*q^3 + 15*q^4 + 30*q^5 + 57*q^6 + 104*q^7 + 183*q^8 + 313*q^9 + ...

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[(1-x^(2*k))^2 * (1-x^(3*k)) * (1-x^(12*k))^3 / ((1-x^k)^3 * (1-x^(4*k)) * (1-x^(6*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

a[n_] := SeriesCoefficient[(EllipticTheta[2, 0, I*q^(3/2)]* EllipticTheta[2, 0, q^3])/(2*EllipticTheta[2, 0, I*q^(1/2)]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[a[n], {n, 50}] (* G. C. Greubel, Nov 27 2017 *)

eta[q_] := q^(1/24)*QPochhammer[q]; A:= eta[q^2]^2*eta[q^3]*eta[q^12]^3/ (eta[q]^3*eta[q^4]*eta[q^6]^2); a:=CoefficientList[Series[A/q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^6 + A)^2), n))}

CROSSREFS

Cf. A186924, A187130, A121373, A000122, A010054, A000700.

Sequence in context: A055795 A058695 A228447 * A209816 A182726 A023610

Adjacent sequences:  A187097 A187098 A187099 * A187101 A187102 A187103

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 05 2011

STATUS

approved

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Last modified December 17 02:32 EST 2018. Contains 318192 sequences. (Running on oeis4.)