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A106402 Expansion of eta(q^3)^9 / eta(q)^3 in powers of q. 6
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 (list; graph; refs; listen; history; text; internal format)
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).

REFERENCES

G. E. Andrews, B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 314, Equ. (14.2.14).

LINKS

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

J. M. Borwein and P. 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.

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

H Movasati, Y Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411, 2016.

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

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.

Convolution cube of A033687.

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

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

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. A033687, A109041, A198958, A231947.

Sequence in context: A285900 A022408 A126827 * A125706 A113510 A163795

Adjacent sequences:  A106399 A106400 A106401 * A106403 A106404 A106405

KEYWORD

nonn,mult

AUTHOR

Michael Somos, May 02 2005

STATUS

approved

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Last modified March 3 04:19 EST 2021. Contains 341756 sequences. (Running on oeis4.)