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

0,2

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

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

Cf. A086614 (triangle), A086615 (antidiagonal sums).

Cf. A006318.

Sequence in context: A371724 A066571 A087648 * A040027 A182968 A071603

Adjacent sequences: A086613 A086614 A086615 * A086617 A086618 A086619

Keyword

nonn

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