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

1, 1, -6, -307, 104742, 251699498, -4229811552588, -497641562809372379, 409828230340907182689774, 2362579011761419853955236859806, -95338580221916838164306067991935130836

Offset: 0

Author

Olivier Gérard

Keywords

• sign

Examples

			G.f. = 1 + x - 6*x^2 - 307*x^3 + 104742*x^4 + 251699498*x^5 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..49
• 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), 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).

Column k=7 of A290789.

Programs

Mathematica

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 *)

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 A015103(n)
  A(-7, n)
end # Seiichi Manyama, Dec 25 2016

Formula

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-7, and a(0) = 1.

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

Extensions

Offset changed to 0 by Seiichi Manyama, Dec 25 2016