A015097 - OEIS

%I #49 Mar 09 2021 19:11:52

%S 1,1,-1,-7,47,873,-26433,-1749159,220526159,56904690761,

%T -29022490524961,-29777360924913095,60924625361199230575,

%U 249669263740090899509545,-2044791574538659983034398465,-33505955988983997787211823466215

%N Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.

%H Seiichi Manyama, <a href="/A015097/b015097.txt">Table of n, a(n) for n = 0..81</a>

%H 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 a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-2 and a(0)=1.

%F G.f: 1/(1-x/(1+2x/(1-4x/(1+8x/(1-16x/(1+... (continued fraction). - _Paul Barry_, Jan 15 2009

%F G.f. satisfies: A(x) = 1 / (1 - x*A(-2*x)). - _Seiichi Manyama_, Dec 27 2016

%e G.f. = 1 + x - x^2 - 7*x^3 + 47*x^4 + 873*x^5 + ...

%t m = 16;

%t ContinuedFractionK[If[i == 1, 1, (-1)^(i+1) 2^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Nov 17 2019 *)

%o (Ruby)

%o def A(q, n)

%o   ary = [1]

%o   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}

%o   ary

%o end

%o def A015097(n)

%o   A(-2, n)

%o end # _Seiichi Manyama_, Dec 24 2016

%o (Python)

%o l=[1]

%o for n in range(1, 21):

%o     l.append(sum([(-2)**i*l[i]*l[n - 1 - i] for i in range(n)]))

%o print(l) # _Indranil Ghosh_, Aug 14 2017

%Y Cf. A227543.

%Y 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), this sequence (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 Column k=2 of A290789.

%K sign

%O 0,4

%A _Olivier Gérard_

%E Offset changed to 0 by _Seiichi Manyama_, Dec 24 2016