A143068 -id:A143068

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A143066

Expansion of phi(x^3) / psi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

+10
4

1, -1, 1, 0, 1, -2, 1, -1, 2, -3, 2, -1, 4, -5, 3, -3, 6, -8, 5, -4, 9, -12, 8, -7, 14, -18, 13, -10, 20, -26, 18, -16, 29, -37, 27, -23, 41, -52, 38, -34, 58, -72, 54, -47, 79, -98, 74, -67, 109, -133, 103, -92, 146, -178, 138, -127, 196, -237, 187, -170, 260

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

OFFSET

0,6

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 10th equation.

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 q^(-1/8) * eta(q) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

Euler transform of period 12 sequence [ -1, 1, 1, 1, -1, -2, -1, 1, 1, 1, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (2/3)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143068.

G.f.: (1 + 2 * x^3 + 2 * x^12 + 2 * x^27 + ...) / (1 + x + x^3 + x^6 + x^10 + ...). [Ramanujan]

G.f.: 1 - x * (1 - x) / (1 - x^4) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^4) * (1 - x^8)) - x^9 * (1 - x) * (1 - x^3) * (1 - x^5) / ((1 - x^4) * (1 - x^8) * (1 - x^12)) + ... [Ramanujan]

-psi6 +2*psi3 -psi1

Expansion of psi(x^3)^2 / (psi(x) * psi(x^6)) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 08 2015

a(n) = A262929(4*n). a(3*n) = A262150(n). a(3*n + 1) = - A262152(n). a(3*n + 2) = A262157(n). - Michael Somos, Nov 08 2015

EXAMPLE

G.f. = 1 - x + x^2 + x^4 - 2*x^5 + x^6 - x^7 + 2*x^8 - 3*x^9 + 2*x^10 + ...

G.f. = 1/q - q^7 + q^15 + q^31 - 2*q^39 + q^47 - q^55 + 2*q^63 - 3*q^71 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {-x^2}, x^2, x], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

a[ n_] := SeriesCoefficient[ 2 x^(1/8) EllipticTheta[ 3, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)

a[ n_] := SeriesCoefficient[ x^(1/8)EllipticTheta[ 2, 0, x^(3/2)]^2 / (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3]), {x, 0, n}]; (* Michael Somos, Nov 08 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

CROSSREFS

Cf. A143068, A262150, A262152, A262157, A262929.

KEYWORD

sign

AUTHOR

Michael Somos, Jul 21 2008

STATUS

approved

A261877

Expansion of psi(x^4) / phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

+10
3

1, 0, 0, 2, 1, 0, 4, 2, 0, 8, 4, 0, 15, 8, 0, 26, 14, 0, 44, 24, 0, 72, 40, 0, 115, 64, 0, 180, 100, 0, 276, 154, 0, 416, 232, 0, 618, 344, 0, 906, 505, 0, 1312, 730, 0, 1880, 1044, 0, 2666, 1480, 0, 3746, 2076, 0, 5220, 2888, 0, 7216, 3988, 0, 9903, 5464, 0

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

OFFSET

0,4

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/2) * eta(q^6) * eta(q^8)^2 / (eta(q^3)^2 * eta(q^4)) in powers of q.

Euler transform of period 24 sequence [ 0, 0, 2, 1, 0, 1, 0, -1, 2, 0, 0, 2, 0, 0, 2, -1, 0, 1, 0, 1, 2, 0, 0, 0, ...].

2 * a(n) = A143068(2*n + 1). a(3*n + 2) = 0.

Convolution inverse is A262929. - Michael Somos, Oct 22 2017

EXAMPLE

G.f. = 1 + 2*x^3 + x^4 + 4*x^6 + 2*x^7 + 8*x^9 + 4*x^10 + 15*x^12 + ...

G.f. = q + 2*q^7 + q^9 + 4*q^13 + 2*q^15 + 8*q^19 + 4*q^21 + 15*q^25 + ...

MATHEMATICA

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

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^8 + A)^2 / (eta(x^3 + A)^2 * eta(x^4 + A)), n))};

CROSSREFS

Cf. A143068, A262929.

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 09 2015

STATUS

approved

A262056

Expansion of phi(q^2) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

+10
1

1, 0, 2, 2, 0, 4, 4, 0, 10, 8, 0, 20, 14, 0, 36, 24, 0, 64, 42, 0, 108, 68, 0, 176, 108, 0, 280, 170, 0, 436, 260, 0, 666, 392, 0, 1000, 584, 0, 1480, 856, 0, 2160, 1240, 0, 3116, 1780, 0, 4448, 2526, 0, 6286, 3552, 0, 8804, 4956, 0, 12228, 6856, 0, 16852

(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).

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 eta(q^4)^5 * eta(q^6) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 24 sequence [ 0, 2, 2, -3, 0, 3, 0, -1, 2, 2, 0, -2, 0, 2, 2, -1, 0, 3, 0, -3, 2, 2, 0, 0, ...].

a(n) = A143068(2*n). a(3*n + 1) = 0.

EXAMPLE

G.f. = 1 + 2*q^2 + 2*q^3 + 4*q^5 + 4*q^6 + 10*q^8 + 8*q^9 + 20*q^11 + ...

MATHEMATICA

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

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 * eta(x^6 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^8 + A)^2), n))};

CROSSREFS

Cf. A143068.

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 09 2015

STATUS

approved

A262938

Expansion of phi(-q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

+10
1

1, -2, 0, 0, 2, 0, 2, -4, 0, -2, 4, 0, 4, -8, 0, -4, 10, 0, 8, -16, 0, -8, 20, 0, 14, -30, 0, -16, 36, 0, 24, -52, 0, -28, 64, 0, 42, -88, 0, -48, 108, 0, 68, -144, 0, -80, 176, 0, 108, -230, 0, -128, 280, 0, 170, -360, 0, -200, 436, 0, 260, -552, 0, -308, 666

(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 eta(q)^2 * eta(q^12) / (eta(q^2) * eta(q^6)^2) in powers of q.

Euler transform of period 12 sequence [ -2, -1, -2, -1, -2, 1, -2, -1, -2, -1, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 6^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A262160.

a(n) = (-1)^n * A143068(n). a(2*n) = A260256(n). a(2*n + 1) = -2 * A261877(n).

a(3*n) = 0.

EXAMPLE

G.f. = 1 - 2*q + 2*q^4 + 2*q^6 - 4*q^7 - 2*q^9 + 4*q^10 + 4*q^12 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A) / (eta(x^2 + A) * eta(x^6 + A)^2), n))};

CROSSREFS

Cf. A143068, A260256, A261877, A262160.

KEYWORD

sign

AUTHOR

Michael Somos, Oct 04 2015

STATUS

approved

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