A000377 Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

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

Offset

0,6

Comments

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

Number 42 of the 74 eta-quotients listed in Table I of Martin (1996).

References

George E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT Press, 1986, p. 260.

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.5).

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

George E. Andrews, Nathan Fine 1916-1994, Notices Amer. Math. Soc., 42 (No. 6, 1995), 678-679.

Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, The Ramanujan Journal, Vol. 20, No. 3 (2009), pp. 375-408; arXiv preprint, arXiv:math/0611300 [math.NT], 2006-2007.

Michael Gilleland, Some Self-Similar Integer Sequences.

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

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

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Fine's Equation.

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

Formula

Expansion of (phi(q) * phi(q^6) + phi(q^2) * phi(q^3)) / 2 = psi(-q^2) * psi(-q^3) * chi(-q^6) * chi(-q^12) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jan 26 2006

Expansion of eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^24)) in powers of q.

Multiplicative with a(0) = 1, a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24). - Michael Somos, Jun 17 2005

Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Jan 26 2006

Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 22 2011

G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(8*k)) / (1 + x^(12*k)).

G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)). - Michael Somos, Sep 10 2005

G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5*n)) / (1 + q^(12*n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007

a(n) = (-1)^n * A190611(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0. a(2*n) = a(3*n) = a(n). a(2*n + 1) = A129402(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.2825... . - Amiram Eldar, Oct 23 2022

Example

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

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -6, #] &]] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] + EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3] QPochhammer[ q^8] QPochhammer[ -q, q] / QPochhammer[ -q^12, q^12] , {q, 0, n}]; (* Michael Somos, May 17 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -6, d)))};
(PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -6, p) * X)))[n])};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^24 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(24), 1), 102); A[1] + A[2] + A[3] + A[4] + A[5] + 2*A[6] + A[7] + 2*A[8] + A[9] + A[10] + 2*A[11] + 2*A[12]; /* Michael Somos, May 17 2015 */

Crossrefs

Cf. A129402, A190611.

Sequence in context: A115660 A128581 A190611 * A192013 A329618 A026517

Adjacent sequences: A000374 A000375 A000376 * A000378 A000379 A000380

Keyword

nonn,easy,nice,mult

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Sep 10 2002

Status

approved