A230535 Expansion of q * (f(-q, -q^7) / f(-q^3, -q^5))^2 in powers of q where f(,) is Ramanujan's two-variable theta function.

1, -2, 1, 2, -4, 4, -1, -6, 11, -8, -1, 12, -20, 16, 2, -22, 34, -30, 1, 40, -64, 52, -2, -68, 113, -88, -2, 112, -180, 144, 2, -182, 284, -228, 4, 286, -452, 356, -4, -440, 698, -544, -5, 668, -1044, 816, 6, -996, 1545, -1210, 6, 1464, -2276, 1768, -7, -2128

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

Euler transform of period 8 sequence [ -2, 0, 2, 0, 2, 0, -2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - u^2) * (1 - v) - 4*u*v.

G.f.: x * (Product_{k>=0} (1 - x^(8*k + 1)) * (1 - x^(8*k + 7)) / ((1 - x^(8*k + 3)) * (1 - x^(8*k + 5))))^2.

a(2*n) = -2 * A224216(n). a(2*n + 1) = A230534(n).

Convolution square of A092869.

Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)). - Simon Plouffe, Mar 02 2021

Example

G.f. = q - 2*q^2 + q^3 + 2*q^4 - 4*q^5 + 4*q^6 - q^7 - 6*q^8 + 11*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 8, k], {k, n}]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2] / (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q, q^8] QPochhammer[ q^7, q^8] / (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8]))^2, {q, 0, n}];

Prog

(PARI) {a(n) = local(A, A2); if( n<2, n==1, n--; A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( (2 * A^2 * A2^2 / (A^2 + A2))^2, n))};

Crossrefs

Cf. A092869, A224216, A230534.

Sequence in context: A295313 A105022 A263293 * A349741 A257651 A275122

Adjacent sequences: A230532 A230533 A230534 * A230536 A230537 A230538

Keyword

sign

Author

Michael Somos, Oct 22 2013

Status

approved