A092876 Expansion of q * (f(-q, -q^12) * f(-q^3, -q^10) * f(-q^4, -q^9)) / (f(-q^2, -q^11) * f(-q^5, -q^8) * f(-q^6, -q^7)) in powers of q where f(, ) is Ramanujan's general theta function.

1, -1, 1, -2, 1, 0, 1, 1, -1, -2, 0, 0, 2, 2, -3, 2, -6, 3, 1, 2, 2, -2, -4, 0, -2, 5, 7, -8, 6, -16, 7, 1, 6, 6, -7, -10, 1, -2, 11, 14, -17, 12, -34, 16, 3, 12, 11, -12, -22, 1, -6, 24, 30, -36, 25, -70, 32, 6, 25, 24, -26, -42, 2, -10, 45, 56, -68, 48, -132, 60, 12, 45, 43, -46, -78, 4, -22, 84, 106, -126, 89, -242, 110, 20, 84, 80

Offset

1,4

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 = 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) = u^3*v * (3 + 3*v + v^2) + 3*u^2*v * (v^2 + v - 1) + u*v * (1 - 3*v + 3*v^2) - (u^4 + v^4)

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

a(n) = A214157(n) - A133099(n) unless n=0. - Michael Somos, Jul 05 2012

Convolution inverse is A214157.

Example

G.f. = q - q^2 + q^3 - 2*q^4 + q^5 + q^7 + q^8 - q^9 - 2*q^10 + 2*q^13 + 2*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[ 13, k], {k, n - 1}], {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)

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))}; /* Michael Somos, Oct 24 2005 */
(PARI) {a(n) = my(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^2 - v + u*v^3 + u^3*v^2 + 2*u*v * (1 - u + v + u*v), k+1) / 2); polcoeff(A, n))};

Crossrefs

Cf. A133099, A214157.

Sequence in context: A321749 A143842 A369460 * A187360 A334368 A370561

Adjacent sequences: A092873 A092874 A092875 * A092877 A092878 A092879

Keyword

sign

Author

Michael Somos, Mar 09 2004

Status

approved