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

0,3

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

Links

Table of n, a(n) for n=0..57.

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Mirror image of A126181.

Cf. A000108, A002212, A026388, A098474.

Sequence in context: A050166 A124959 A081281 * A371686 A321000 A121289

Adjacent sequences: A108195 A108196 A108197 * A108199 A108200 A108201

Keyword

nonn,tabl,easy

Author

Emeric Deutsch, Jun 15 2005, Mar 30 2007

Extensions

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

Status

approved