OFFSET
0,2
COMMENTS
Row sums of triangle A086614. - Paul D. Hanna, Jul 24 2003
Hankel transform is A136577(n+1). - Paul Barry, Jun 03 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), #09.7.6.
FORMULA
G.f.: A(x) = 1/(1 - x)^2 + x*A(x)^2.
a(1) = 1 and a(n) = n + Sum_{i=1..n-1} a(i)*a(n-i) for n >= 2. - Benoit Cloitre, Mar 16 2004
G.f.: (1 - x - sqrt(1 - 6*x + x^2))/(2*x*(1 - x)). Cf. A001003. - Ralf Stephan, Mar 23 2004
a(n) = Sum_{k=0..n} C(n+k+1, 2*k+1) * A000108(k). - Paul Barry, Jun 03 2009
Recurrence: (n+1)*a(n) = (7*n-2)*a(n-1) - (7*n-5)*a(n-2) + (n-2)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
A(x) = 1/(1 - x)^2 * c(x/(1-x^2)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024
EXAMPLE
a(1) = 2 + 1 = 3;
a(2) = 3 + 4 + 2 = 9;
a(3) = 4 + 10 + 12 + 5 = 31;
a(4) = 5 + 20 + 42 + 40 + 14 = 121.
MATHEMATICA
Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])/(2*x*(1-x)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
PROG
(Sage) # Generalized algorithm of L. Seidel
def A086616_list(n) :
D = [0]*(n+2); D[1] = 1
b = True; h = 2; R = []
for i in range(2*n) :
if b :
for k in range(h, 0, -1) : D[k] += D[k-1]
else :
for k in range(1, h, 1) : D[k] += D[k-1]
R.append(D[h-1]); h += 1;
b = not b
return R
A086616_list(23) # Peter Luschny, Jun 02 2012
(PARI) x='x+O('x^66); Vec((1-x-sqrt(1-6*x+x^2))/(2*x*(1-x))) \\ Joerg Arndt, May 10 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Jul 24 2003
EXTENSIONS
Name changed using a comment of Emeric Deutsch from Dec 20 2004. - Peter Luschny, Jun 03 2012
STATUS
approved