login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A106640 Row sums of A059346. 6
1, 1, 4, 11, 36, 117, 393, 1339, 4630, 16193, 57201, 203799, 731602, 2643903, 9611748, 35130195, 129018798, 475907913, 1762457595, 6550726731, 24428808690, 91377474411, 342763939656, 1289070060903, 4859587760076, 18360668311027, 69514565858653, 263693929034909 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = p(n + 1) where p(x) is the unique degree-n polynomial such that p(k) = Catalan(k) for k = 0, 1, ..., n. - Michael Somos, Jan 05 2012

Number of Dyck (n+1)-paths whose minimum ascent length is 1. - David Scambler, Aug 22 2012

a(n) is the number of ordered rooted trees with n+2 nodes such that the minimal outdegree equals 1. a(2) = 4:

.  o    o      o      o

.  |    |     / \    / \

.  o    o    o   o  o   o

.  |   / \   |          |

.  o  o   o  o          o

.  |

.  o                        - Alois P. Heinz, Jun 29 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: (sqrt( 1 - 2*x - 3*x^2 ) / (1 + x) - sqrt( 1 - 4*x )) / (2*x^2) = 2 / (sqrt( 1 - 2*x - 3*x^2 ) + (1 + x) * sqrt( 1 - 4*x )). - Michael Somos, Jan 05 2012

a(n) = A000108(n+1) - A005043(n+1).

a(n) ~ 2^(2*n+2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 21 2017

EXAMPLE

1 + x + 4*x^2 + 11*x^3 + 36*x^4 + 117*x^5 + 393*x^6 + 1339*x^7 + 4630*x^8 + ...

a(2) = 4 since p(x) = (x^2 - x + 2) / 2 interpolates p(0) = 1, p(1) = 1, p(2) = 2, and p(3) = 4. - Michael Somos, Jan 05 2012

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1],

      ((30*n^3-44*n^2-22*n+24)*a(n-1)-(25*n^3-105*n^2+140*n-48)*a(n-2)

       -6*(n-1)*(5*n-4)*(2*n-3)*a(n-3))/(n*(n+2)*(5*n-9)))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Jun 29 2014

MATHEMATICA

max = 30; t = Table[Differences[Table[CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; a[n_] := Sum[t[[n-k+1, k]], {k, 1, n}]; Array[a, max] (* Jean-François Alcover, Jan 21 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, n++; subst( polinterpolate( vector(n, k, binomial( 2*k - 2, k - 1) / k)), x, n + 1))} /* Michael Somos, Jan 05 2012 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 / (sqrt( 1 - 2*x - 3*x^2 + A) + (1 + x) * sqrt( 1 - 4*x + A)) , n))} /* Michael Somos, Jan 05 2012 */

CROSSREFS

Cf. A000108, A005043, A244455.

Sequence in context: A149237 A054577 A206687 * A109268 A256960 A174993

Adjacent sequences:  A106637 A106638 A106639 * A106641 A106642 A106643

KEYWORD

nonn

AUTHOR

Philippe Deléham, May 26 2005

EXTENSIONS

Typo in a(20) corrected and more terms from Alois P. Heinz, Jun 29 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 07:17 EDT 2018. Contains 316378 sequences. (Running on oeis4.)