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A086616 Partial sums of the large Schroeder numbers (A006318). 5
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

P 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, a(n)=n+sum(i=1, n-1, a(i)*a(n-i)). - 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 February 20 19:04 EST 2018. Contains 299381 sequences. (Running on oeis4.)