A214157 Expansion of (1/x) * (f(-x^2, -x^11) * f(-x^5, -x^8) * f(-x^6, -x^7)) / (f(-x, -x^12) * f(-x^3, -x^10) * f(-x^4, -x^9)) in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 0, 1, 2, 0, -1, 0, -1, -1, 0, -2, 0, 4, 1, -2, 3, 4, -2, -3, -1, -2, -2, -2, -5, 0, 9, 3, -4, 8, 12, -4, -7, -1, -6, -6, -4, -12, -1, 22, 6, -10, 17, 24, -9, -16, -3, -12, -11, -8, -25, -1, 45, 14, -20, 36, 52, -18, -32, -6, -25, -24, -16, -50, -2, 88

Offset

-1,5

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 13 sequence [1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u*v^3 - u^3*v^2 + 2*u*v * (1 + u - v + u*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^3 - u) * (u^3 - v) - 3*u*v * (1 + u + v) * (u*v - u - v).

G.f.: (1/x) * Product_{k>0} (1 - x^k)^-Kronecker(13, k).

a(n) = A092876(n) + A133099(n) unless n=0.

Convolution inverse of A092876.

Example

G.f. = 1/x + 1 + x^2 + 2*x^3 - x^5 - x^7 - x^8 - 2*x^10 + 4*x^12 + x^13 - 2*x^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/x Product[ (1 - x^k) ^ -KroneckerSymbol[ 13, k], {k, n+1}], {x, 0, n}];

Prog

(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod( k=1, n, (1 - x^k)^-kronecker( 13, k), 1 + x * O(x^n)), n))};

Crossrefs

Cf. A092876, A133099.

Sequence in context: A235924 A097304 A136745 * A246720 A343030 A246690

Adjacent sequences: A214154 A214155 A214156 * A214158 A214159 A214160

Keyword

sign

Author

Michael Somos, Jul 05 2012

Status

approved