A085261 Expansion of chi(x) / phi(x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

1, 1, -2, -1, 5, 3, -9, -5, 18, 10, -30, -16, 53, 29, -85, -44, 139, 73, -215, -110, 335, 172, -502, -253, 755, 382, -1104, -550, 1614, 805, -2312, -1142, 3305, 1631, -4650, -2277, 6525, 3193, -9041, -4395, 12486, 6063, -17070, -8247, 23255, 11218, -31414, -15090, 42289, 20285

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

George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), #R1.

M. Ishikawa and J. Zeng, The Andrews-Stanley partition function and Al-Salam-Chihara polynomials, Disc. Math., 309 (2009), 151-175. (See t(n) p. 151. Note that there is a typo in the g.f. for f(n) - see A144558.) [Added by N. J. A. Sloane, Jan 25 2009.]

O. P. Lossers, Comparing Odd Parts in Conjugate Partitions: Solution 10969, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.

Michael Somos, Introduction to Ramanujan theta functions

R. P. Stanley, Problem 10969, Amer. Math. Monthly, 109 (2002), 760.

Formula

Expansion of psi(x) / f(x^2)^2 in powers of x where psi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2014

Expansion of q^(1/24) * eta(q^2)^4 * eta(q^8)^2 / (eta(q) * eta(q^4)^6) in powers of q.

Euler transform of period 8 sequence [1, -3, 1, 3, 1, -3, 1, 1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246712. - Michael Somos, Sep 02 2014

G.f.: Product_{k>0} (1 + x^(2*k - 1)) / ((1 - x^(4*k)) * (1 + x^(4*k - 2))^2).

Example

G.f. = 1 + x - 2*x^2 - x^3 + 5*x^4 + 3*x^5 - 9*x^6 - 5*x^7 + 18*x^8 + 10*x^9 - ...
G.f. = 1/q + q^23 - 2*q^47 - q^71 + 5*q^95 + 3*q^119 - 9*q^143 - 5*q^167 + 18*q^191 + ...

Maple

t1:=mul( (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))^2), n=1..100): t2:=series(t1, q, 100): f:=n->coeff(t2, q, n); # N. J. A. Sloane, Jan 25 2009

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / EllipticTheta[ 3, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 01 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8) QPochhammer[ -x^2]^2), {x, 0, n}]; (* Michael Somos, Sep 02 2014 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / eta(x + A) / eta(x^4 + A)^6, n))};
(PARI) {a(n) = polcoeff( prod( k=1, ( n+1)\2, 1 + x^(2*k - 1), 1 + x * O(x^n)) / prod(k=1, (n+2)\4, (1 - x^(4*k)) * (1 + x^(4*k - 2))^2, 1 + x * O(x^n)), n)};

Crossrefs

Cf. A000041, A097566, A190101, A246712.

Sequence in context: A175330 A333259 A334172 * A179218 A131119 A114901

Adjacent sequences: A085258 A085259 A085260 * A085262 A085263 A085264

Keyword

sign

Author

Michael Somos, Jun 23 2003

Status

approved