OFFSET
0,3
COMMENTS
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 chi(-x) * phi(x^2) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 2, 0, -3, -1, 1, -1, -1, 0, 2, -1, -4, -1, 2, 0, -1, -1, 1, -1, -3, 0, 2, -1, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
a(2*n) = A261115(n).
a(2*n + 1) = - A263548(n).
a(8*n + 4) = a(8*n + 6) = 0.
a(4*n + 1) = -a(n).
a(4*n + 3) = -2 * A128582(n).
a(8*n) = A113780(n).
a(8*n + 2) = 2 * A260089(n).
a(16*n + 3) = -2 * A128583(n).
a(16*n + 7) = -2 * A128591(n).
EXAMPLE
G.f. = 1 - x + 2*x^2 - 2*x^3 - x^5 - 2*x^7 + 3*x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 - 2*q^10 - q^16 - 2*q^22 + 3*q^25 - 2*q^28 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 2, #] KroneckerSymbol[ -3, m/#] &]]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ x^6] QPochhammer[ x, x^6] QPochhammer[ x^5, x^6], {x, 0, n}];
PROG
(PARI) {a(n) = my(m); if( n<0, 0, m = 3*n + 1; (-1)^n * sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A)^2), n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor(3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, (-1)^e, p%24==1 || p%24==7, e+1, p%24==5 || p%24==11, (e+1)*(-1)^e, !(e%2))))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Dec 23 2016
STATUS
approved