A015097 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.

1, 1, -1, -7, 47, 873, -26433, -1749159, 220526159, 56904690761, -29022490524961, -29777360924913095, 60924625361199230575, 249669263740090899509545, -2044791574538659983034398465, -33505955988983997787211823466215

Offset: 0

Author

Olivier Gérard

Keywords

• sign

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..81
• Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

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), 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).

Column k=2 of A290789.

Programs

Mathematica

m = 16;
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 *)

Python

l=[1]
for n in range(1, 21):
    l.append(sum([(-2)**i*l[i]*l[n - 1 - i] for i in range(n)]))
print(l) # Indranil Ghosh, Aug 14 2017

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 A015097(n)
  A(-2, n)
end # Seiichi Manyama, Dec 24 2016

Formula

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

G.f: 1/(1-x/(1+2x/(1-4x/(1+8x/(1-16x/(1+... (continued fraction). - Paul Barry, Jan 15 2009

G.f. satisfies: A(x) = 1 / (1 - x*A(-2*x)). - Seiichi Manyama, Dec 27 2016

Extensions

Offset changed to 0 by Seiichi Manyama, Dec 24 2016