A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 12, 12, 14, 12, 12, 16, 16, 18, 18, 16, 18, 24, 24, 24, 21, 28, 27, 24, 28, 24, 30, 32, 36, 32, 24, 36, 38, 36, 42, 32, 40, 36, 42, 48, 36, 48, 48, 48, 43, 42, 48, 56, 52, 54, 48, 48, 54, 56, 60, 48, 62, 60, 54, 64, 56, 72, 66, 64, 72, 48, 72, 72

Offset

1,2

Comments

Number 24 of the 74 eta-quotients listed in Table I of Martin (1996).

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..5000

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^3) / eta(q))^2 * eta(q^4) * eta(q^12) in powers of q.

Euler transform of period 12 sequence [ 2, 0, 0, -1, 2, -2, 2, -1, 0, 0, 2, -4, ...].

a(n) is multiplicative with a(p^e) = p^e if p<5, a(p^e) = (p^(e+1) - 1) / (p-1) if p == 1, 11 (mod 12), a(p^e) = (p^(e+1) + (-1)^e) / (p+1) if p == 5, 7 (mod 12).

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 - x^(2*k) + x^(4*k)).

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(4*k)) * (1 - x^(12*k)).

a(2*n) = 2 * a(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(6*sqrt(3)) = 0.949703... (A258414). - Amiram Eldar, Dec 22 2023

Example

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

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ n/d KroneckerSymbol[ 12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^2 QPochhammer[ q^4] QPochhammer[ q^12]/QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 12, d)))};
(PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 12, p); (p^(e+1) - f^(e+1)) / (p - f)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A) / eta(x + A)^2, n))};

Crossrefs

Cf. A258414.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A102443 A102441 A102440 * A081328 A179276 A213635

Adjacent sequences: A124812 A124813 A124814 * A124816 A124817 A124818

Keyword

nonn,easy,mult

Author

Michael Somos, Nov 08 2006

Status

approved