A086616 Partial sums of the large Schroeder numbers (A006318).
1, 3, 9, 31, 121, 515, 2321, 10879, 52465, 258563, 1296281, 6589727, 33887465, 175966211, 921353249, 4858956287, 25786112993, 137604139011, 737922992937, 3974647310111, 21493266631001, 116642921832963, 635074797251889, 3467998148181631, 18989465797056721, 104239408386028035
Offset: 0
Examples
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.
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.
Programs
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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 *) -
PARI
x='x+O('x^66); Vec((1-x-sqrt(1-6*x+x^2))/(2*x*(1-x))) \\ Joerg Arndt, May 10 2013 -
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
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
Extensions
Name changed using a comment of Emeric Deutsch from Dec 20 2004. - Peter Luschny, Jun 03 2012
Comments