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A015086 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=5. 23
1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, 48995209411107768186, 19138851672289046707772366, 37380607950584029444762130426196 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..53

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=5 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(5*x)) = 1/(1-x/(1-5*x/(1-5^2*x/(1-5^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 26 2016

EXAMPLE

G.f. = 1 + x + 6*x^2 + 161*x^3 + 20466*x^4 + 12833546*x^5 + 40130703276*x^6 + ...

MATHEMATICA

a[n_] := a[n] = Sum[5^i*a[i]*a[n -i -1], {i, 0, n -1}];

a[0] = 1; Array[a, 12, 0] (* Robert G. Wilson v, Dec 24 2016 *)

m = 11; ContinuedFractionK[If[i == 1, 1, -5^(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 A015086(n)

  A(5, 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), this sequence (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).

Column k=5 of A090182, A290759.

Sequence in context: A241453 A193370 A332116 * A052466 A280477 A078535

Adjacent sequences:  A015083 A015084 A015085 * A015087 A015088 A015089

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

Offset changed to 0 by Seiichi Manyama, Dec 24 2016

STATUS

approved

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Last modified April 13 03:43 EDT 2021. Contains 342934 sequences. (Running on oeis4.)