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 A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function. 37
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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). 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 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions 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. EXAMPLE G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ... 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 *) PROG (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 (MAGMA) A := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* Michael Somos, Jun 10 2014 */ 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.) Sequence in context: A145425 A201909 A070352 * A239678 A136374 A303869 Adjacent sequences:  A227213 A227214 A227215 * A227217 A227218 A227219 KEYWORD sign AUTHOR Michael Somos, Sep 21 2013 STATUS approved

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Last modified October 25 06:05 EDT 2020. Contains 338011 sequences. (Running on oeis4.)