A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

1, 3, 4, 2, 1, 3, 6, 4, 0, -1, 4, 6, 4, 2, 2, 2, 3, 4, 2, 0, 1, 6, 8, 2, 0, 3, 6, 0, -2, 0, 6, 6, 4, 4, 2, 4, 3, 4, 0, -2, 0, 6, 8, 2, 2, -1, 6, 4, 2, 1, 4, 6, 4, 2, 0, 6, 0, 0, 0, 0, 4, 6, 8, 2, 1, 2, 12, 4, -2, -2, 2, 6, 0, 2, 2, 2, 0, 8, 4, 0, 3, 3, 8, 2

Offset: 0

Author

Michael Somos, Sep 21 2013

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Zagier (2009) refers to Case D corresponding to the Apery numbers (A005258).

Examples

			G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ...
		

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.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• D. Zagier, Integral solutions of Apery-like recurrence equations.

Crossrefs

Cf. A005258, A078905, A229802.

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.)

Programs

Magma

A := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* Michael Somos, Jun 10 2014 */

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ Re[(3 - I) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)

PARI

{a(n) = if( n<1, n==0, sumdiv(n, d, real( (3 - I) * [ 0, 1, I, -I, -1][ d%5 + 1])))};

PARI

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -3, 2, 2, -3][k%5 + 1], 1 + x * O(x^n)), n))};

Sage

A = ModularForms( Gamma1(5), 1, prec=20) . basis(); A[0] + 3*A[1]; # Michael Somos, Jun 10 2014

Formula

Expansion of f(-q)^2 * (f(-q^5) / f(-q, -q^4))^5 = f(-q^2, -q^3)^2 * (f(-q^5) / f(-q, -q^4))^3 in powers of q where f() is a Ramanujan theta function.

Euler transform of period 5 sequence [ 3, -2, -2, 3, -2, ...].

Moebius transform is period 5 sequence [ 3, 1, -1, -3, 0, ...]. - Michael Somos, Jun 10 2014

G.f. = g(t(q)) where g(), t() are the g.f. for A005258 and A078905.

G.f.: (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)))^5.