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.

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

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 A358914 A189994

Adjacent sequences: A193880 A193881 A193882 * A193884 A193885 A193886

Keyword

nonn

Author

Michael Somos, Aug 07 2011

Status

approved