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A029552 Expansion of phi(x) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. 6
1, 3, 4, 7, 13, 19, 29, 43, 62, 90, 126, 174, 239, 325, 435, 580, 769, 1007, 1313, 1702, 2191, 2808, 3580, 4539, 5735, 7216, 9036, 11278, 14028, 17383, 21474, 26448, 32471, 39759, 48550, 59123, 71829, 87053, 105249, 126975, 152858, 183623 (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

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/24) * eta(q^2)^5 /(eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 17 2004

Euler transform of period 4 sequence [ 3, -2, 3, 0, ...]. - Michael Somos, Sep 17 2004

G.f. A(x) is the limit of x^(n^2) P_{2n}(1/x) where P_n(q) = Sum_{k=0..n} C(n,k;q) and C(n,k;q) is q-binomial coefficients. See A083906 for P_n. - Michael Somos, Sep 17 2004

G.f.: (1 + 2 * Sum_{k>0} x^(k^2)) / (Product_{k>0} (1 - x^k)).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(7/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017

EXAMPLE

G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 13*x^4 + 19*x^5 + 29*x^6 + 43*x^7 + ...

G.f. = 1/q + 3*q^23 + 4*q^47 + 7*q^71 + 13*q^95 + 19*q^119 + 29*q^143 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^2 / QPochhammer[ q, q^2], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 17 2004 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^2), n)); } /* Michael Somos, Sep 17 2004 */

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, 2*n, prod(i=1, k, (1 -x^(2*n + 1-i)) / (1 - x^i))), n^2-n))}; /* Michael Somos, Sep 17 2004 */

CROSSREFS

Cf. A083906, A098613.

Sequence in context: A093124 A055664 A089374 * A193883 A227038 A189994

Adjacent sequences:  A029549 A029550 A029551 * A029553 A029554 A029555

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 30 06:59 EDT 2020. Contains 333119 sequences. (Running on oeis4.)