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A086616 Partial sums of the large Schroeder numbers (A006318). 7
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 (list; graph; refs; listen; history; text; internal format)
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: A151037 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

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Last modified April 17 10:38 EDT 2021. Contains 343064 sequences. (Running on oeis4.)