A328798 -id:A328798

Search: a328798 -id:a328798

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A330311

Where n appears in A329544, or -1 if n never appears.

+10
4

1, 3, 2, 5, 4, 9, 13, 12, 45, 40, 7, 56, 18, 11, 14, 15, 10, 19, 6, 44, 43, 8, 55, 17, 28, 21, 16, 25, 20, 29, 63, 42, 23, 175, 34, 27, 22, 31, 26, 35, 30, 80, 75, 24, 39, 33, 48, 37, 32, 49, 36, 85, 79, 76, 46, 187, 130, 47, 38, 51, 50, 89, 84, 110, 109

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

OFFSET

1,2

COMMENTS

a(919) is unknown. If it is not -1, it is greater than 10^6 (see A329544).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..918

FORMULA

a(A104444(n)) = -1. - Rémy Sigrist, Dec 11 2019

CROSSREFS

Cf. A104444, A329544, A329797, A328798.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 10 2019

STATUS

approved

A328795

Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

+10
3

1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 6, 2, 0, 3, 6, 4, 0, 4, 8, 4, 0, 7, 14, 7, 0, 6, 16, 10, 0, 11, 20, 11, 0, 14, 32, 16, 0, 17, 38, 21, 0, 22, 46, 24, 0, 32, 66, 34, 0, 34, 78, 44, 0, 49, 96, 50, 0, 60, 130, 66, 0, 72, 154, 84, 0, 90, 186, 98, 0, 117, 244

(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).

Convolution square of A328802.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.

LINKS

Table of n, a(n) for n=0..73.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).

a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

EXAMPLE

G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...

G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A112206, A328789, A328790, A328797, A328798, A328802.

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 28 2019

STATUS

approved

A328797

Expansion of (chi(-x) * chi(x^3))^2 in powers of x where chi() is a Ramanujan theta function.

+10
3

1, -2, 1, 0, 0, -2, 2, 0, 2, -2, 1, 0, 2, -6, 2, 0, 3, -6, 4, 0, 4, -8, 4, 0, 7, -14, 7, 0, 6, -16, 10, 0, 11, -20, 11, 0, 14, -32, 16, 0, 17, -38, 21, 0, 22, -46, 24, 0, 32, -66, 34, 0, 34, -78, 44, 0, 49, -96, 50, 0, 60, -130, 66, 0, 72, -154, 84, 0, 90, -186

(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).

Convolution square of A328800.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328790.

LINKS

Table of n, a(n) for n=0..69.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

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

a(n) = (-1)^n * A328795(n). a(2*n) = A112206(n).

a(4*n) = A328789(n). a(4*n + 1) = -2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

EXAMPLE

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

G.f. = q^-1 - 2*q^2 + q^5 - 2*q^14 + 2*q^17 + 2*q^23 - 2*q^26 + ..

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A112206, A328789, A328790, A328795, A328798, A328800.

KEYWORD

sign

AUTHOR

Michael Somos, Oct 27 2019

STATUS

approved

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