A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.

1, 3, 9, 13, 24, 27, 50, 51, 81, 72, 120, 117, 170, 150, 216, 205, 288, 243, 362, 312, 450, 360, 528, 459, 601, 510, 729, 650, 840, 648, 962, 819, 1080, 864, 1200, 1053, 1370, 1086, 1530, 1224, 1680, 1350, 1850, 1560, 1944, 1584, 2208, 1845, 2451, 1803, 2592

Offset

1,2

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Number 3 of the 74 eta-quotients listed in Table I of Martin (1996).

a(n+1) is the number of partition triples of n where each partition is 3-core (see Theorem 3.1 of Wang link).

Convolution cube of A033687.

Convolution square is A198958. - Michael Somos, Dec 26 2015

References

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vaclav Kotesovec)

Jonathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012). See page 697.

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

Hossein Movasati and Younes Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2021.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Formula

Expansion of (c(q) / 3)^3 in powers of q where c(q) is a cubic AGM theta function.

Euler transform of period 3 sequence [ 3, 3, -6, ...].

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.

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.

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

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.

a(3*n) = 9 * a(n). a(3*n + 1) = A231947(n). - Michael Somos, May 18 2015

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

Example

G.f. = q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, #^2 KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3]^3 / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
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 *)

Prog

(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))};
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( n/d, 3)))};
(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)))};
(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
(Magma) A := Basis( ModularForms( Gamma1(3), 3), 52); A[2]; /* Michael Somos, May 18 2015 */

Crossrefs

Cf. A004016, A005882, A005928, A033687, A109041, A129404, A198958, A231947.

Sequence in context: A022408 A126827 A363628 * A125706 A113510 A163795

Adjacent sequences: A106399 A106400 A106401 * A106403 A106404 A106405

Keyword

nonn,easy,mult

Author

Michael Somos, May 02 2005

Status

approved