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A214262
Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.
36
1, -5, 9, -11, 24, -45, 50, -53, 81, -120, 120, -99, 170, -250, 216, -203, 288, -405, 362, -264, 450, -600, 528, -477, 601, -850, 729, -550, 840, -1080, 962, -821, 1080, -1440, 1200, -891, 1370, -1810, 1530, -1272, 1680, -2250, 1850, -1320, 1944, -2640, 2208
OFFSET
1,2
COMMENTS
Zagier (2009) writes "... associated to the weight 3 Eisenstein series g(z) = Sigma b(n)q^n = q - 5q^2 + 9q^3 - 11q^4 + ...".
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
REFERENCES
D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.
FORMULA
Expansion of (1/9) * c(q) * b(q)^2 * c(q^2) / b(q^2) = (c(q)^3 - 8*c(q^2)^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - Michael Somos, Oct 06 2013
a(n) is multiplicative with a(3^e) = 9^e, a(2^e) = -(4^(e+1) + 9*(-1)^(e+1)) / 5, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111661.
G.f.: Sum_{k>0} -(-1)^k * k^2 * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker( -3, k) * (x^k - x^(2*k)) / (1 + x^k)^3.
EXAMPLE
G.f. = q - 5*q^2 + 9*q^3 - 11*q^4 + 24*q^5 - 45*q^6 + 50*q^7 - 53*q^8 + 81*q^9 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 06 2013 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^5 QPochhammer[ q^3] QPochhammer[ q^6]^4 / QPochhammer[ q^2]^4, {q, 0, n}]; (* Michael Somos, Oct 06 2013 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* Michael Somos, Oct 06 2013 */
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^3 + A) * eta(x^6 + A)^4 / eta(x^2 + A)^4, n))};
(PARI) {a(n) = my(A, p, e); if( n<0, 0, A = factor( n); prod( k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if( p==3, 9^e, if( p==2, -(4^(e+1) + 9*(-1)^(e+1)) / 5, if( p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1)))))))};
CROSSREFS
Cf. A111661.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
Sequence in context: A314604 A309137 A190894 * A288143 A120228 A053749
KEYWORD
sign,mult
AUTHOR
Michael Somos, Jul 09 2012
STATUS
approved