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%I #63 Nov 09 2023 08:53:02
%S 1,3,9,13,24,27,50,51,81,72,120,117,170,150,216,205,288,243,362,312,
%T 450,360,528,459,601,510,729,650,840,648,962,819,1080,864,1200,1053,
%U 1370,1086,1530,1224,1680,1350,1850,1560,1944,1584,2208,1845,2451,1803,2592
%N Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C Number 3 of the 74 eta-quotients listed in Table I of Martin (1996).
%C a(n+1) is the number of partition triples of n where each partition is 3-core (see Theorem 3.1 of Wang link).
%C Convolution cube of A033687.
%C Convolution square is A198958. - _Michael Somos_, Dec 26 2015
%D George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001). See p. 314, Eq. (14.2.14).
%H Seiichi Manyama, <a href="/A106402/b106402.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vaclav Kotesovec)
%H Jonathan M. Borwein and Peter B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012). See page 697.
%H Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Hossein Movasati and Younes Nikdelan, <a href="http://arxiv.org/abs/1603.09411">Gauss-Manin Connection in Disguise: Dwork Family</a>, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%H Liuquan Wang, <a href="http://arxiv.org/abs/1507.03099">Explicit Formulas for Partition Pairs and Triples with 3-Cores</a>, arXiv:1507.03099 [math.NT], 2015.
%F Expansion of (c(q) / 3)^3 in powers of q where c(q) is a cubic AGM theta function.
%F Euler transform of period 3 sequence [ 3, 3, -6, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 6*u*v*w + 8*u*w^2 - u^2*w.
%F G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) = x * Product_{k>0} (1 - x^(3*k))^9 / (1 - x^k)^3.
%F a(n) is multiplicative and a(p^e) = ((p^2)^(e+1) - u^(e+1)) / (p^2 - u) where u = 0, 1, -1 when p == 0, 1, 2 (mod 3). - _Michael Somos_, Oct 19 2005
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 27^(-1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109041.
%F a(3*n) = 9 * a(n). a(3*n + 1) = A231947(n). - _Michael Somos_, May 18 2015
%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 4*Pi^3/(81*sqrt(3)) = 0.8840238... (A129404). - _Amiram Eldar_, Nov 09 2023
%e G.f. = q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, #^2 KroneckerSymbol[ n/#, 3] &]]; (* _Michael Somos_, Jul 19 2012 *)
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3]^3 / QPochhammer[ q])^3, {q, 0, n}]; (* _Michael Somos_, Jul 19 2012 *)
%t nmax = 40; Rest[CoefficientList[Series[x * Product[(1 - x^(3*k))^9 / (1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^9 / eta(x + A)^3, n))};
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( n/d, 3)))};
%o (PARI) {a(n) = my(A, p, e, u); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; u = kronecker(-3, p); ((p^2)^(e+1) - u^(e+1)) / (p^2 - u)))};
%o (PARI) a(n) = sumdiv(n, d, ((d % 3) == 1)*(n/d)^2) - sumdiv(n, d, ((d % 3)== 2)*(n/d)^2); \\ _Michel Marcus_, Jul 14 2015
%o (Magma) A := Basis( ModularForms( Gamma1(3), 3), 52); A[2]; /* _Michael Somos_, May 18 2015 */
%Y Cf. A004016, A005882, A005928, A033687, A109041, A129404, A198958, A231947.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, May 02 2005
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