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A163746 Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q. 6
1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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. 82, Eq. (32.53).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

M. Somos, Introduction to Ramanujan theta functions

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).

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

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

Cf. A122856, A122865, A125061, A138741.

Sequence in context: A138952 A138950 A125061 * A004591 A195588 A153510

Adjacent sequences:  A163743 A163744 A163745 * A163747 A163748 A163749

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Aug 03 2009

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.