A015098 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-3.

1, 1, -2, -23, 586, 48778, -11759396, -8596478231, 18783386191762, 123275424165263086, -2426183754235085042972, -143268577734839493464012630, 25379312219817753259837452498340

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..65

Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Formula

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-3 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-3*x)) = 1/(1-x/(1+3*x/(1-3^2*x/(1+3^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Example

G.f. = 1 + x - 2*x^2 - 23*x^3 + 586*x^4 + 48778*x^5 + ...

Mathematica

a[1] := 1; a[n_] := a[n] = Sum[(-3)^(i - 1)*a[i]*a[n - i], {i, 1, n - 1}]; Array[a, 20, 1] (* G. C. Greubel, Dec 24 2016 *)
m = 13; ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

Prog

(Ruby)
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015098(n)
  A(-3, n)
end # Seiichi Manyama, Dec 24 2016

Crossrefs

Cf. A227543.

Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), this sequence (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).

Column k=3 of A290789.

Sequence in context: A090314 A084322 A073062 * A269021 A136039 A237580

Adjacent sequences: A015095 A015096 A015097 * A015099 A015100 A015101

Keyword

sign

Author

Olivier Gérard

Extensions

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

Status

approved