cp's OEIS Frontend

A015103 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-7.

%I A015103 #36 Aug 04 2024 16:30:44
%S A015103 1,1,-6,-307,104742,251699498,-4229811552588,-497641562809372379,
%T A015103 409828230340907182689774,2362579011761419853955236859806,
%U A015103 -95338580221916838164306067991935130836
%N A015103 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-7.
%H A015103 Seiichi Manyama, <a href="/A015103/b015103.txt">Table of n, a(n) for n = 0..49</a>
%H A015103 Robin Sulzgruber, <a href="https://doi.org/10.25365/thesis.30616">The Symmetry of the q,t-Catalan Numbers</a>, Thesis, University of Vienna, 2013.
%F A015103 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-7, and a(0) = 1.
%F A015103 G.f. satisfies: A(x) = 1 / (1 - x*A(-7*x)) = 1/(1-x/(1+7*x/(1-7^2*x/(1+7^3*x/(1-...))))) (continued fraction). - _Seiichi Manyama_, Dec 28 2016
%e A015103 G.f. = 1 + x - 6*x^2 - 307*x^3 + 104742*x^4 + 251699498*x^5 + ...
%t A015103 m = 11; ContinuedFractionK[If[i == 1, 1, -(-7)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)
%o A015103 (Ruby)
%o A015103 def A(q, n)
%o A015103   ary = [1]
%o A015103   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
%o A015103   ary
%o A015103 end
%o A015103 def A015103(n)
%o A015103   A(-7, n)
%o A015103 end # _Seiichi Manyama_, Dec 25 2016
%Y A015103 Cf. A227543.
%Y A015103 Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), this sequence (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), A015095 (q=10), A015096 (q=11).
%Y A015103 Column k=7 of A290789.
%K A015103 sign
%O A015103 0,3
%A A015103 _Olivier Gérard_
%E A015103 Offset changed to 0 by _Seiichi Manyama_, Dec 25 2016