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%I #34 Nov 24 2023 12:42:01
%S 1,2,2,2,1,2,2,2,3,0,2,2,2,2,0,4,2,2,2,0,1,2,4,2,0,2,2,2,3,2,2,0,2,2,
%T 0,2,4,2,2,0,2,4,0,4,0,2,2,2,1,0,4,2,2,0,2,2,2,4,2,0,3,2,2,2,0,0,2,4,
%U 2,0,2,4,2,2,0,0,2,2,4,2,4,2,0,2,0,4,0
%N Expansion of s(12)^3*s(18)^2/(s(6)^2*s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.
%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.38).
%H Seiichi Manyama, <a href="/A097195/b097195.txt">Table of n, a(n) for n = 0..10000</a>
%F Fine gives an explicit formula for a(n) in terms of the divisors of n.
%F a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
%F From _Michael Somos_, Nov 03 2005: (Start)
%F G.f.: Sum_{k} x^k / (1 - x^(6*k + 1)).
%F G.f.: Sum_{k>=0} a(k) * x^(6*k + 1) = Sum_{k>0} x^(2*k-1) * (1 - x^(4*k - 2)) * (1 - x^(8*k - 4)) * (1 - x^(20*k - 10)) / (1 - x^(36*k - 18)). (End)
%F From _Michael Somos_, Mar 05 2016: (Start)
%F Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
%F Euler transform of period 6 sequence [ 2, -1, 0, -1, 2, -2, ...].
%F 6 * a(n) = A004016(6*n + 1). (End)
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - _Amiram Eldar_, Nov 24 2023
%e G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
%e G.f. = q + 2*q^7 + 2*q^13 + 2*q^19 + q^25 + 2*q^31 + 2*q^37 + 2*q^43 + ...
%t a[n_] := DivisorSum[6n+1, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 100} ] (* _Jean-François Alcover_, Nov 23 2015, after _Michael Somos_ *)
%t QP = QPochhammer; s = QP[q^2]^3*(QP[q^3]^2/QP[q]^2/QP[q^6]) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%t a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(6 n + 1))]; (* _Michael Somos_, Mar 05 2016 *)
%o (PARI) {a(n) = if( n<0, 0, sumdiv(6*n+1, d, kronecker(-3, d)))}; /* _Michael Somos_, Nov 03 2005 */
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 6*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p>3, if( p%6==1, e+1, !(e%2)))))}; /* _Michael Somos_, Nov 03 2005 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* _Michael Somos_, Nov 03 2005 */
%Y cf. A004016, A010815, A093602.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Sep 16 2004
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