OFFSET
1,3
COMMENTS
REFERENCES
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) - 1 in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) - 1 in powers of q.
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1+(-1)^e)/2 if p == 3 (mod 4). [corrected by Amiram Eldar, Nov 14 2023]
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
a(n) = A125061(n) unless n=0. a(12*n + 7) = a(12*n + 11) = 0.
a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). - Michael Somos, Sep 02 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 14 2023
EXAMPLE
G.f. = q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + 3 EllipticTheta[ 3, 0, q^3]^2) / 4 - 1, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
(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==2, 1, p==3, e%2*2 + 1, p%4==1, e+1, 1-e%2)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^2) - 1, n))};
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Aug 03 2009
STATUS
approved