A131429 -id:A131429 - cp's OEIS frontend

A131430 Row sums of triangle A131429.

1, 2, 7, 25, 88, 311, 1114, 4050, 14917, 55528, 208459, 787920, 2994655, 11433998, 43824437, 168520201, 649840740, 2512011359, 9731179138, 37768570827, 146833956266, 571711568905, 2229035221824, 8701426599353, 34005503702176, 133030452858279, 520905732440782

Offset: 0

Gary W. Adamson, Jul 10 2007

nonn

			a(3) = 25 = sum of row 3 terms of triangle A131429: (5 + 5 + 6 + 9).

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000108, A131427, A131428, A131429.

a(n)={binomial(2*n,n) - n - 1 + sum(k=0, n, binomial(2*k,k)/(k+1))} \\ Andrew Howroyd, Aug 28 2018

a(n) = binomial(2*n, n) - (n+1) + Sum_{k=0..n} binomial(2*k, k)/(k+1). - Andrew Howroyd, Aug 28 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 28 2018

A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

Original entry on oeis.org

1, 1, 3, 9, 27, 83, 263, 857, 2859, 9723, 33591, 117571, 416023, 1485799, 5348879, 19389689, 70715339, 259289579, 955277399, 3534526379, 13128240839, 48932534039, 182965127279, 686119227299, 2579808294647, 9723892802903, 36734706144303, 139067101832007

Offset: 0

Gary W. Adamson, Jul 10 2007

nonn

Starting (1, 3, 9, 27, 83, ...), = row sums of triangle A136522. - Gary W. Adamson, Jan 02 2008
Hankel transform is A171552. - Paul Barry, Dec 11 2009
Apparently, for n >= 1, the maximum peak height minus the maximum valley height summed over all Dyck n-paths (with max valley height deemed zero if no valleys). - David Scambler, Oct 05 2012
Apparently for n > 1 the number of fixed points in all Dyck (n-1)-paths. A fixed point occurs when a vertex of a Dyck k-path is also a vertex of the path U^kD^k. - David Scambler, May 01 2013

			a(3) = 9 = 2*C(3) - 1 = 2*5 - 1, where C refers to the Catalan numbers, A000108.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A131427, A131429.
Cf. A131428, A118976, A136522.
Cf. A080233, A156644.

List([0..25], n-> 2*Binomial(2*n,n)/(n+1) - 1); # G. C. Greubel, Aug 12 2019

[2*Catalan(n) -1: n in [0..25]]; // G. C. Greubel, Aug 12 2019

seq(2*binomial(2*n,n)/(n+1)-1, n=0..25); # Emeric Deutsch, Jul 25 2007

2CatalanNumber[Range[0,25]]-1  (* Harvey P. Dale, Apr 17 2011 *)

vector(25, n, n--; 2*binomial(2*n,n)/(n+1) - 1) \\ G. C. Greubel, Aug 12 2019

[2*catalan_number(n) -1 for n in (0..25)] # G. C. Greubel, Aug 12 2019

Right border of triangle A131429.
From Emeric Deutsch, Jul 25 2007: (Start)
a(n) = 2*binomial(2*n,n)/(n+1) - 1.
G.f.: (1-sqrt(1-4*x))/x - 1/(1-x). (End)
(1, 3, 9, 27, 83, ...) = row sums of A118976. - Gary W. Adamson, Aug 31 2007
Row sums of triangle A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007
Starting with offset 1 = Narayana transform (A001263) of [1,2,2,2,...]. - Gary W. Adamson, Jul 29 2011
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +3*(-n+1)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^2 = Sum_{k=0..n} A080233(n,k)^2 = Sum_{k=0..n} A156644(n,k)^2. - Seiichi Manyama, Mar 25 2025

More terms from Emeric Deutsch, Jul 25 2007

A131427 A000108(n) preceded by n zeros.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 0, 429, 0, 0, 0, 0, 0, 0, 0, 0, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4862, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 58786, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 208012

Offset: 0

Gary W. Adamson, Jul 10 2007

Triangle given by A000004 DELTA A000012 where DELTA is the operator defined in A084938. - Philippe Deléham, Jul 12 2007

T(n,k) is the number of Dyck paths of semilength n having exactly k U=(1,1) steps. - Alois P. Heinz, Jun 09 2014

			First few rows of the triangle are:
1;
0, 1;
0, 0, 2;
0, 0, 0, 5;
0, 0, 0, 0, 14;
0, 0, 0, 0, 0, 42;
...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A131428, A131429, A000108, A243752.

T:= (n, k)-> `if`(kAlois P. Heinz, Jun 09 2014

T[n_, n_] := CatalanNumber[n]; T[, ] = 0;
Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 20 2016 *)

A000108(n) preceded by n zeros, as an infinite lower triangular matrix.

More terms from Philippe Deléham, Oct 16 2008

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A131430 Row sums of triangle A131429.

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A131428 a(n) = 2*C(n) - 1, where C(n) = A000108(n) are the Catalan numbers.

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A131427 A000108(n) preceded by n zeros.

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