A108198 Triangle read by rows: T(n,k) = binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0 <= k <= n).

1, 1, 2, 1, 4, 5, 1, 6, 15, 14, 1, 8, 30, 56, 42, 1, 10, 50, 140, 210, 132, 1, 12, 75, 280, 630, 792, 429, 1, 14, 105, 490, 1470, 2772, 3003, 1430, 1, 16, 140, 784, 2940, 7392, 12012, 11440, 4862, 1, 18, 180, 1176, 5292, 16632, 36036, 51480, 43758, 16796, 1, 20, 225

Offset: 0

Author

Emeric Deutsch, Jun 15 2005, Mar 30 2007

Keywords

• nonn
• tabl
• easy

Comments

Also, with offset 1, triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and ending at the point (2k,0) (1 <= k <= n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. For example, T(3,2)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDDL.

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 1 for the Catalan sequence A000108. See A098474 for a similar comment. - Wolfdieter Lang, Jun 25 2019

Examples

			Triangle begins:
1;
1,  2;
1,  4,  5;
1,  6, 15,  14;
1,  8, 30,  56,  42;
1, 10, 50, 140, 210, 132;
1, 12, 75, 280, 630, 792, 429;
		

Links

• E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Crossrefs

Mirror image of A126181.

Cf. A000108, A002212, A026388, A098474.

Programs

Maple

T:=(n,k)->binomial(2*k+2,k+1)*binomial(n,k)/(k+2): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
seq(print(seq(4^k*coeff(h(n), x, k), k=0..n)), n=0..9); # Peter Luschny, Feb 03 2015

Mathematica

Flatten[Table[Binomial[2k+2,k+1] Binomial[n,k]/(k+2),{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 20 2013 *)

Sage

def A108198(n,k):
    return (-1)^k*catalan_number(k+1)*rising_factorial(-n,k)/factorial(k)
for n in range(7): [A108198(n,k) for k in (0..n)] # Peter Luschny, Feb 05 2015

Formula

Sum of row n = A002212(n+1).

T(n,n) = Catalan(n+1) (A000108).

Sum_{k=1..n} k*T(n,k) = A026388(n).

With offset 1, T(n,k) = c(k)*binomial(n-1,k-1), where c(j) = binomial(2j,j)/(j+1) is a Catalan number (A000108).

G.f.: G-1, where G=G(t,z) satisfies G = 1 + t*z*G^2 + z*(G-1).

T(n, k) = 4^k*[x^k]hypergeometric([3/2, -n], [3], -x). - Peter Luschny, Feb 03 2015, based on an observation of Peter Bala in A254632.

T(n, k) = (-1)^k*Catalan(k+1)*Pochhammer(-n,k)/k!. - Peter Luschny, Feb 05 2015

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew Plewe, Jun 16 2007