A124233 Expansion of psi(q) * phi(-q^10) * chi(-q^5) / chi(-q^2) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2

Offset

0,4

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 38 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of eta(q^2) * eta(q^4) * eta(q^5) * eta(q^10) / (eta(q) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [ 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, ...].

Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].

a(n) is multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20).

G.f.: 1 + Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)).

a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.

a(n) = A035170(n) unless n=0. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Dec 22 2023

Example

G.f. = 1 + q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -20, #] &]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^5] QPochhammer[ q^10] / (QPochhammer[ q] QPochhammer[ q^20]), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -20, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^10 + A) / eta(x + A) / eta(x^20 + A), n))};

Crossrefs

Cf. A033764, A035170, A129390.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A066888 A029313 A144001 * A035170 A111949 A143323

Adjacent sequences: A124230 A124231 A124232 * A124234 A124235 A124236

Keyword

nonn,easy,mult

Author

Michael Somos, Oct 21 2006

Status

approved