A113417 Expansion of phi(x) * phi(-x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, -2, -4, 8, 7, -10, -12, 8, 18, -18, -16, 24, 21, -20, -28, 32, 20, -32, -36, 24, 42, -42, -28, 48, 57, -36, -52, 40, 36, -58, -60, 56, 48, -66, -48, 72, 74, -42, -80, 80, 61, -82, -72, 56, 90, -96, -64, 72, 98, -70, -100, 104, 64, -106, -108, 72, 114, -96

Offset

0,2

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * (eta(q) * eta(q^8))^2 * (eta(q^2) / eta(q^4))^3 in powers of q.

Euler transform of period 8 sequence [ -2, -5, -2, -2, -2, -5, -2, -4, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).

G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (2*k - 1) * (-1)^[k/2] * x^(2*k - 1) / (1 - x^(4*k - 2)) = x * (Product_{k>0} ((1 - x^(2*k)) * (1 - x^(4*k)) * (1 + x^(8*k)))^2 / (1 + x^(4*k))).

a(n) = (-1)^n * A113419(n) = (-1)^floor(n/2) * A209940(n) = (-1)^(n + floor(n/2)) * A258096(n). - Michael Somos, May 19 2015

a(n) = A117000(2*n + 1).

a(n) = Sum_{d | 2*n + 1} Kronecker(2, d) * d.

Example

G.f. = 1 - 2*x - 4*x^2 + 8*x^3 + 7*x^4 - 10*x^5 - 12*x^6 + 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 - 4*q^5 + 8*q^7 + 7*q^9 - 10*q^11 - 12*q^13 + 8*q^15 + ...

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^8]^2 QPochhammer[ x^2]^3 / QPochhammer[ x^4]^3, {x, 0, n}]; (* Michael Somos, May 19 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, d * (d%2) * (-1)^((d + 1) \ 4)))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p *= kronecker( 2, p); (p^(e+1) - 1) / (p - 1))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^3, n))};
(Magma) A := Basis( ModularForms( Gamma1(16), 2), 117); A[2] - 2*A[4] - 4*A[6] + 8*A[8] + 7*A[10] - 10*A[12] - 12*A[14]; /* Michael Somos, May 19 2015 */

Crossrefs

Cf. A113419, A117000, A209940, A258096.

Sequence in context: A195715 A343629 A039794 * A113419 A209940 A258096

Adjacent sequences: A113414 A113415 A113416 * A113418 A113419 A113420

Keyword

sign

Author

Michael Somos, Oct 29 2005

Status

approved