A015095 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=10.

1, 1, 11, 1121, 1123331, 11235577641, 1123580257785051, 1123582505161487376561, 11235827298801257861061293171, 1123582752351801734250808539216885881

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..45

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=10 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(10*x)) = 1/(1-x/(1-10*x/(1-10^2*x/(1-10^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 27 2016

Example

G.f. = 1 + x + 11*x^2 + 1121*x^3 + 1123331*x^4 + 11235577641*x^5 + ...

Mathematica

a[n_] := a[n] = Sum[10^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
m = 10; ContinuedFractionK[If[i == 1, 1, -10^(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 A015095(n)
  A(10, 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), A015098 (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), this sequence (q=10), A015096 (q=11).

Column k=10 of A090182, A290759.

Sequence in context: A303067 A075600 A078568 * A167118 A045981 A090507

Adjacent sequences: A015092 A015093 A015094 * A015096 A015097 A015098

Keyword

nonn

Author

Olivier Gérard

Extensions

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

Status

approved