A260649 Expansion of (phi(q^3) * phi(q^5) + phi(q) * phi(q^15)) / 2 - 1 in powers of q where phi(q) is a Ramanujan theta function.

1, 0, 1, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 3, 2, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 2, 0, 2, 0, 0

Offset

1,8

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 = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q^2)^2 * eta(q^6) * eta(q^10) * eta(q^30)^2) / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60)) - 1 in powers of q.

a(n) is multiplicative with a(2^e) = |e-1|, a(3^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1 + (-1)^e)/2 if p == 7, 11, 13, 14 (mod 15).

Moebius transform of a period 60 sequence.

G.f.: Sum_{k>0} Kronecker(-15, k) x^k / (1 - (-x)^k).

a(n) = A122855(n) unless n=0.

a(3*n) = a(5*n) = a(n). a(4*n) = A035175(n). a(4*n + 2) = 0.

a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.

Example

G.f. = x + x^3 + x^4 + x^5 + 2*x^8 + x^9 + x^12 + x^15 + 3*x^16 + 2*x^17 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -15, #] If[ Mod[#, 4] == 2, -1, 1] &]];
a[ n_] := If[ n < 1, 0, Times@@ (Which[# == 1, 1, # == 2, #2 - 1, # < 6, 1, KroneckerSymbol[#, -15] == 1, #2 + 1, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[QPochhammer[ q^2]^2 QPochhammer[ q^6] QPochhammer[ q^10] QPochhammer[ q^30]^2 / (QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^15] QPochhammer[ q^60]) - 1, {q, 0, n}];

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-15, d) * (-1)^(d%4==2) ))};
(PARI) {a(n) = if( n<1, 0, qfrep( [1, 0; 0, 15], n)[n] + qfrep( [3, 0; 0, 5], n)[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==2, e-1, p==3 || p==5, 1, kronecker(p, -15) == 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)^2 * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A)) - 1, n))};

Crossrefs

Cf. A035175, A122855.

Sequence in context: A253242 A359814 A359815 * A122855 A140727 A140728

Adjacent sequences: A260646 A260647 A260648 * A260650 A260651 A260652

Keyword

nonn,mult

Author

Michael Somos, Nov 12 2015

Status

approved