|
|
A015097
|
|
Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-2.
|
|
26
|
|
|
1, 1, -1, -7, 47, 873, -26433, -1749159, 220526159, 56904690761, -29022490524961, -29777360924913095, 60924625361199230575, 249669263740090899509545, -2044791574538659983034398465, -33505955988983997787211823466215
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
LINKS
|
|
|
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
|
|
EXAMPLE
|
G.f. = 1 + x - x^2 - 7*x^3 + 47*x^4 + 873*x^5 + ...
|
|
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 *)
|
|
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
A(-2, n)
(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)]))
|
|
CROSSREFS
|
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).
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|