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A085261 Expansion of chi(x) / phi(x^2) in powers of x where phi(), chi() are Ramanujan theta functions. 4
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 (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).

REFERENCES

M. Ishikawa and J. Zeng, The Andrews-Stanley partition function ..., Disc. Math., 309 (2009), 151-175. (See f(n) on p. 151, but 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, Solution 10969, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.

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

LINKS

Table of n, a(n) for n=0..49.

G. E. Andrews, On a Partition Function of Richard Stanley

M. Somos, Introduction to Ramanujan theta functions

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: A026205 A082748 A175330 * A179218 A131119 A114901

Adjacent sequences:  A085258 A085259 A085260 * A085262 A085263 A085264

KEYWORD

sign

AUTHOR

Michael Somos, Jun 23 2003

STATUS

approved

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Last modified October 18 16:38 EDT 2017. Contains 293524 sequences.