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%I #63 Oct 23 2022 02:54:48
%S 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,
%T 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,
%U 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
%N Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 42 of the 74 eta-quotients listed in Table I of Martin (1996).
%D George E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT Press, 1986, p. 260.
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.5).
%H T. D. Noe, <a href="/A000377/b000377.txt">Table of n, a(n) for n = 0..10000</a>
%H George E. Andrews, <a href="http://www.ams.org/notices/199506/fine.pdf">Nathan Fine 1916-1994</a>, Notices Amer. Math. Soc., 42 (No. 6, 1995), 678-679.
%H Alexander Berkovich and Hamza Yesilyurt, <a href="https://doi.org/10.1007/s11139-009-9215-8">Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms</a>, The Ramanujan Journal, Vol. 20, No. 3 (2009), pp. 375-408; <a href="http://arXiv.org/abs/math.NT/0611300">arXiv preprint</a>, arXiv:math/0611300 [math.NT], 2006-2007.
%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>.
%H Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FinesEquation.html">Fine's Equation</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F 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
%F Expansion of eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^24)) in powers of q.
%F 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
%F 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
%F 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, ...].
%F 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
%F G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(8*k)) / (1 + x^(12*k)).
%F 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
%F G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5*n)) / (1 + q^(12*n)) (see Berkovich/Yesilyurt). - _Ralf Stephan_, May 14 2007
%F 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).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.2825... . - _Amiram Eldar_, Oct 23 2022
%e 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 + ...
%t a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -6, #] &]] (* _Michael Somos_, Jul 11 2011 *)
%t 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 *)
%t 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 *)
%o (PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -6, d)))};
%o (PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -6, p) * X)))[n])};
%o (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))};
%o (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 */
%Y Cf. A129402, A190611.
%K nonn,easy,nice,mult
%O 0,6
%A _N. J. A. Sloane_
%E Edited by _Michael Somos_, Sep 10 2002
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