login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A015092
Carlitz-Riordan q-Catalan numbers (recurrence version) for q=8.
23
1, 1, 9, 593, 304857, 1249312673, 40939981188777, 10732252327798007281, 22507185898866512901924729, 377607964391970470904956530918721, 50681683810611444451901001718927186370889
OFFSET
0,3
FORMULA
a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=8 and a(0)=1.
G.f. satisfies: A(x) = 1 / (1 - x*A(8*x)) = 1/(1-x/(1-8*x/(1-8^2*x/(1-8^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 26 2016
EXAMPLE
G.f. = 1 + x + 9*x^2 + 593*x^3 + 304857*x^4 + 1249312673*x^5 + ...
MATHEMATICA
a[n_] := a[n] = Sum[8^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 = 11; ContinuedFractionK[If[i == 1, 1, -8^(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 A015092(n)
A(8, 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), this sequence (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=8 of A090182, A290759.
Sequence in context: A067320 A061611 A238609 * A332159 A139107 A226552
KEYWORD
nonn
EXTENSIONS
Offset changed to 0 by Seiichi Manyama, Dec 24 2016
STATUS
approved