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A193883 Expansion of x * phi(x) * psi(x^14) / (f(-x) * f(-x^7)) in powers of x where phi(), psi(), f() are Ramanujan theta functions. 2
0, 1, 3, 4, 7, 13, 19, 29, 44, 65, 94, 133, 187, 258, 354, 481, 651, 871, 1154, 1526, 1998, 2603, 3376, 4358, 5594, 7148, 9103, 11531, 14560, 18320, 22972, 28708, 35757, 44413, 54990, 67904, 83626, 102736, 125890, 153882, 187694, 228396, 277336 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of chi(-x^2)^2 / (chi(-x)^3 * chi(-x^7) * chi(-x^14)^4 ) in powers of x where chi() is a Ramanujan theta function.

Expansion of q^(-5/12) * eta(q^2)^5 * eta(q^28)^2 / (eta(q)^3 * eta(q^4)^2 * eta(q^7) * eta(q^14)) in powers of q.

Euler transform of period 28 sequence [ 3, -2, 3, 0, 3, -2, 4, 0, 3, -2, 3, 0, 3, 0, 3, 0, 3, -2, 3, 0, 4, -2, 3, 0, 3, -2, 3, 0, ...].

a(n) = A102314(4*n + 2).

a(n) ~ exp(4*Pi*sqrt(n/21)) / (2^(5/2) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

EXAMPLE

G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 19*x^6 + 29*x^7 + 44*x^8 + 65*x^9 + ...

G.f. = q^17 + 3*q^29 + 4*q^41 + 7*q^53 + 13*q^65 + 19*q^77 + 29*q^89 + 44*q^101 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, 4 n + 2}];

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4]^2 / (QPochhammer[ x^7, x^14] QPochhammer[ x^14, x^28]^2 QPochhammer[ x, x^2]^3), {x, 0, n - 1}];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^7] / (2 QPochhammer[ x] QPochhammer[ x^7]), {x, 0, n + 3/4}];

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^28 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^7 + A) * eta(x^14 + A)), n))};

CROSSREFS

Cf. A102314.

Sequence in context: A055664 A089374 A029552 * A227038 A189994 A125118

Adjacent sequences:  A193880 A193881 A193882 * A193884 A193885 A193886

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 07 2011

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)