A182033 -id:A182033

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A230256

Expansion of f(-x) * psi(x^2) * phi(x^3) / f(-x^3)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

+10
6

1, -1, 0, 4, -6, 1, 11, -19, 4, 31, -50, 11, 77, -122, 28, 173, -273, 62, 370, -573, 130, 751, -1149, 261, 1461, -2214, 498, 2750, -4125, 923, 5022, -7472, 1663, 8936, -13202, 2919, 15551, -22817, 5019, 26521, -38681, 8467, 44417, -64438, 14035, 73197

(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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/12) * eta(q) * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^5 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ -1, 0, 4, -2, -1, 0, -1, -2, 4, 0, -1, 0, ...].

a(n) = A132179(2*n) = A062242(4*n) = A062244(4*n) = A132301(4*n) = A182056(4*n) = A182036(6*n) = A182032(12*n - 1).

a(n) = A058531(12*n) = A093073(12*n) = A132976(12*n) = A143840(12*n) = A164268(12*n) = A164612(12*n) = A182033(12*n) = A193261(12*n). - Michael Somos, Jan 29 2015

EXAMPLE

G.f. = 1 - x + 4*x^3 - 6*x^4 + x^5 + 11*x^6 - 19*x^7 + 4*x^8 + 31*x^9 + ...

G.f. = q^-1 - q^11 + 4*q^35 - 6*q^47 + q^59 + 11*q^71 - 19*q^83 + 4*q^95 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 3, 0, x^3] QPochhammer[ x] / (2 x^(1/4) QPochhammer[ x^3]^3), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^2), n))};

CROSSREFS

Cf. A062242, A062244, A132179, A132301, A182032, A182036, A182056.

Cf. A058531, A093073, A132976, A143840, A164268, A164612, A182033, A193261, A254372.

KEYWORD

sign

AUTHOR

Michael Somos, Oct 14 2013

STATUS

approved

A122830

Expansion of c(q) * c(q^6) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

+10
3

1, 1, 0, -2, -3, 0, 5, 7, 0, -12, -15, 0, 26, 32, 0, -50, -63, 0, 92, 114, 0, -168, -201, 0, 295, 350, 0, -496, -591, 0, 818, 967, 0, -1332, -1554, 0, 2126, 2468, 0, -3324, -3855, 0, 5126, 5916, 0, -7824, -8970, 0, 11793, 13471, 0, -17548, -20007, 0, 25857, 29384, 0, -37788, -42771, 0

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

OFFSET

1,4

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^3 / (eta(q) * eta(q^6)^7) in powers of q.

Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 4*v - 4*u^2 + 4*v^2 + 2*w^2 + 8*u*v + 8*v*w + 18*u*v*w + 3*u*w^2 - 12*u^2*w - 12*u^2*v + 6*v^2*w - 3*v^3 - 9*u^2*w^2 - 18*u^2*v*w - 9*u*v^2*w - 9*u^2*v^2 - 9*v^3*w.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u6*u1 - 2*u3*u2 + u6*u2^2 - 3*u6*u3*u2 - 3*u3^2*u2 - 4*u3*u2^2 - 3*u3^2*u2^2 + 6*u3^2*u1 - 4*u3*u2*u1 + 4*u6*u2*u1 + u6*u1^2 + 2*u3*u1^2 + 6*u3^2*u1^2 + 3*u6*u3*u1^2 - 6*u6*u3*u2^2 + 6*u3^2*u2*u1 - 6*u6*u3*u2*u1.

Convolution inverse is A182033. - Michael Somos, Feb 19 2015

a(3*n) = 0. - Michael Somos, Feb 19 2015

EXAMPLE

G.f. = q + q^2 - 2*q^4 - 3*q^5 + 5*q^7 + 7*q^8 - 12*q^10 - 15*q^11 + ...

MATHEMATICA

eta[x_] := x^(1/24)*QPochhammer[x]; A122830[n_] := SeriesCoefficient[

eta[q^2]^2* eta[q^3]^3*eta[q^18]^3/(eta[q]*eta[q^6]^7 ), {q, 0, n}]; Table[A122830[n], {n, 0, 50}] (* G. C. Greubel, Aug 11 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^3 / (eta(x + A) * eta(x^6 + A)^7), n))};

CROSSREFS

Cf. A182033.

KEYWORD

sign

AUTHOR

Michael Somos, Sep 12 2006

STATUS

approved

A233034

Expansion of (f(-x^2) / phi(-x^3))^2 in powers of x where phi(), f() are Ramanujan theta functions.

+10
3

1, 0, -2, 4, -1, -8, 14, -4, -23, 40, -10, -60, 98, -24, -140, 224, -54, -304, 478, -112, -627, 968, -224, -1236, 1884, -432, -2346, 3540, -801, -4320, 6454, -1448, -7742, 11472, -2556, -13548, 19936, -4408, -23226, 33952, -7462, -39080, 56800, -12416, -64660

(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).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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^(-2/3) * b(q^2) * c(q^2) / (3 * f(-q^3)^4) in powers of q where b(), c() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [ 0, -2, 4, -2, 0, 0, ...].

G.f.: Product_{k>0} ( (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(3*k))^2 )^2.

a(n) = A092848(2*n) = A128111(2*n) = A182057(4*n) = A062242(4*n + 1) = A182056(4*n + 1) = A139032(6*n + 1) = A164615(6*n + 1) = A182033(6*n + 1) = A058531(12*n + 2) = A093073(12*n + 2) = A128143(12*n + 2) = A128145(12*n + 2) = A143840(12*n + 2) = A182032(12*n + 2) = A193261(12*n + 2).

-a(n) = A062244(4*n + 1) = A182034(6*n + 1) = A182035(6*n + 1) = A128144(12*n + 2) = A132976(12*n + 3) = A164268(12*n + 2) = A164612(12*n + 3) = A182035(12*n + 2).

EXAMPLE

G.f. = 1 - 2*x^2 + 4*x^3 - x^4 - 8*x^5 + 14*x^6 - 4*x^7 - 23*x^8 + 40*x^9 + ...

G.f. = q - 2*q^13 + 4*q^19 - q^25 - 8*q^31 + 14*q^37 - 4*q^43 - 23*q^49 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ x^3]^2)^2, {x, 0, n}];

PROG

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

CROSSREFS

Cf. A058531, A062242, A062244, A092848, A093073, A128111, A128143, A128144, A128145, A132976, A139032, A143840, A164268, A164612, A164615, A182032, A182033, A182034, A182035, A182056, A182057, A193261.

KEYWORD

sign

AUTHOR

Michael Somos, Dec 03 2013

STATUS

approved

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