A210066 -id:A210066

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A134746

Expansion of 1+k in powers of q^(1/2) where q is Jacobi's nome and k is the elliptic modulus.

+10
4

1, 4, 0, -16, 0, 56, 0, -160, 0, 404, 0, -944, 0, 2072, 0, -4320, 0, 8648, 0, -16720, 0, 31360, 0, -57312, 0, 102364, 0, -179104, 0, 307672, 0, -519808, 0, 864960, 0, -1419456, 0, 2299832, 0, -3682400, 0, 5831784, 0, -9141808, 0, 14194200, 0, -21842368, 0

(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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (phi(q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Expansion of (eta(q^8) / eta(q))^4 * (eta(q^2) / eta(q^4))^14 in powers of q.

Euler transform of period 8 sequence [ 4, -10, 4, 4, 4, -10, 4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A210066.

G.f.: ( (Sum_{k in Z} x^(k^2)) / (Sum_{k in Z} x^(2*k^2)) )^2 = ( Product_{k>0} (1 + x^k)^2 * (1 + x^(4*k))^2 / (1 + x^(2*k))^5 )^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v)^2 - u * (2 - u) * v^2.

a(2*n) = 0 unless n=0. a(2*n + 1) = 4 * A001938(n) = A127393(n).

a(n) = (-1)^n * A210067(n). Convolution inverse of A210066. - Michael Somos, Oct 16 2015

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 34 + 24*sqrt(2) - 4*sqrt(140 + 99*sqrt(2)). - Simon Plouffe, Mar 04 2021

EXAMPLE

G.f. = 1 + 4*q - 16*q^3 + 56*q^5 - 160*q^7 + 404*q^9 - 944*q^11 + 2072*q^13 + ...

MATHEMATICA

CoefficientList[Series[(QPochhammer[x^8]/QPochhammer[x])^4 (QPochhammer[x^2]/QPochhammer[x^4])^14, {x, 0, 50}], x] (* Jan Mangaldan, Mar 21 2013 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x^8 + A) / eta(x + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^7 )^2, n))};

CROSSREFS

Cf. A001938, A127393, A210066, A210067.

KEYWORD

sign

AUTHOR

Michael Somos, Nov 07 2007

STATUS

approved

A260145

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

+10
1

1, -4, 12, -32, 78, -176, 376, -768, 1509, -2872, 5316, -9600, 16966, -29408, 50088, -83968, 138738, -226196, 364284, -580032, 913824, -1425552, 2203368, -3376128, 5130999, -7738136, 11585208, -17225472, 25444278, -37350816, 54504160, -79085568, 114133296

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5)^2 in powers of q.

Euler transform of period 8 sequence [ -4, 6, -4, 4, -4, 6, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A210067.

G.f.: x * Product_{k>0} ( 1 + x^(2*k))^6 * (1 + x^(4*k))^4 / (1 + x^k)^4.

a(n) = -(-1)^n * A107035(n). -4 * a(n) = A210066(n) unless n=0. -8 * a(n) = A139820(n) unless n=0.

a(2*n) = -4 * A092877(n). a(2*n + 1) = A022577(n). a(4*n) = -32 * A014103(n).

Convolution square of A210063. Convolution inverse of A131125.

a(n) ~ -(-1)^n * exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

EXAMPLE

G.f. = x - 4*x^2 + 12*x^3 - 32*x^4 + 78*x^5 - 176*x^6 + 376*x^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^2]^2 / EllipticTheta[ 3, 0, q]^2, {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5)^2, n))};