A132004 Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, -1, 1, -1, 2, -1, 0, -1, 1, -2, 0, -1, 2, 0, 2, -1, 2, -1, 0, -2, 0, 0, 0, -1, 3, -2, 1, 0, 2, -2, 0, -1, 0, -2, 0, -1, 2, 0, 2, -2, 2, 0, 0, 0, 2, 0, 0, -1, 1, -3, 2, -2, 2, -1, 0, 0, 0, -2, 0, -2, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 3, 0, 0, -2

Offset

1,5

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

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

Expansion of (1 - eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.

a(n) is multiplicative with a(2^e) = 2*0^e - 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).

G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(-36, k).

a(3*n) = a(n). -2 * a(n) = A132003(n) unless n = 0. a(2*n) = - A035154(n). a(2*n + 1) = A125079(n).

a(n) = (-1)^n * A035154(n). a(12*n + 7) = a(12*n + 11) = 0. - Michael Somos, Nov 01 2015

a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 2) = - A125079(n). - Michael Somos, Nov 01 2015

a(6*n) = - A035154(n). a(6*n + 2) = - A122865(n). a(6*n + 4) = - A122856(n). - Michael Somos, Nov 01 2015

a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). - Michael Somos, Nov 01 2015

Example

G.f. = x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n+d) * kronecker( -36, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Crossrefs

Cf. A008441, A035154, A053692, A122856, A122865, A125079, A132003, A258277, A258278.

Sequence in context: A035188 A342148 A066295 * A035154 A113446 A121450

Adjacent sequences: A132001 A132002 A132003 * A132005 A132006 A132007

Keyword

sign,mult

Author

Michael Somos, Aug 06 2007

Status

approved