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A015085 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=4. 23
1, 1, 5, 89, 5885, 1518897, 1558435125, 6386478643785, 104648850228298925, 6858476391221411106209, 1797922152786660462507074405, 1885261615172756172119161342909753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..58

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=4 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(4*x)) = 1/(1-x/(1-4*x/(1-4^2*x/(1-4^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 26 2016

EXAMPLE

G.f. = 1 + x + 5*x^2 + 89*x^3 + 5885*x^4 + 1518897*x^5 + 1558435125*x^6 + ...

From Seiichi Manyama, Dec 05 2016: (Start)

a(1) = 1,

a(2) = 4^1 + 1 = 5,

a(3) = 4^3 + 4^2 + 2*4^1 + 1 = 89,

a(4) = 4^6 + 4^5 + 2*4^4 + 3*4^3 + 3*4^2 + 3*4^1 + 1 = 5885. (End)

MATHEMATICA

a[n_] := a[n] = Sum[4^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 = 12; ContinuedFractionK[If[i == 1, 1, -4^(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 A015085(n)

  A(4, 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), this sequence (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=4 of A090182, A290759.

Sequence in context: A330605 A028353 A191512 * A258181 A067258 A167424

Adjacent sequences:  A015082 A015083 A015084 * A015086 A015087 A015088

KEYWORD

nonn

AUTHOR

Olivier Gérard

EXTENSIONS

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

STATUS

approved

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Last modified August 2 15:26 EDT 2021. Contains 346428 sequences. (Running on oeis4.)