A071948 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

1, 1, 2, 1, 4, 7, 1, 6, 18, 30, 1, 8, 33, 88, 143, 1, 10, 52, 182, 455, 728, 1, 12, 75, 320, 1020, 2448, 3876, 1, 14, 102, 510, 1938, 5814, 13566, 21318, 1, 16, 133, 760, 3325, 11704, 33649, 76912, 120175, 1, 18, 168, 1078, 5313, 21252, 70840, 197340, 444015

Offset: 0

Author

N. J. A. Sloane, Jun 15 2002

Keywords

• nonn
• easy
• tabl

Comments

This is the table of h(n,k) in the notation of Carlitz (p.125). The triangle (with an offset of 1 rather than 0) enumerates two-line arrays of positive integers

............1 a_2 ... a_(n-1) a_n..........

............1 b_2 ... b_(n-1) b_n..........

such that a_i <= i (2 <= i <= n) and b_2 <= a_2 <= ... <= b_n <= a_n = k.

See A193091 and A211788 for other two-line array enumerations. - Peter Bala, Aug 02 2012

Examples

			Triangle begins
  1;
  1, 2;
  1, 4,  7;
  1, 6, 18, 30;
  1, 8, 33, 88, 143;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
• Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
• L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
• S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
• P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
• M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Crossrefs

Row sums give A001764.

Rows are the reversals of the rows of A092276.

Cf. A193091, A211788.

Programs

Maple

T := proc(n,k) if k<=n then (n-k+1)*binomial(2*n+k+1,k)/(n+1) else 0 fi end: seq(seq(T(n,k),k=0..n),n=0..10);

Mathematica

t[n_, k_] /; k <= n := (n-k+1)*Binomial[2*n+k+1, k]/(n+1); t[, ] = 0; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Sage

# Computes the first n rows of the triangle.
def A071948_triangle(n) :
    D = [0 for i in (0..n+1)]; D[1] = 1
    for i in (4..2*n+3) :
        h = i//2 - 1
        for k in (1..h) : D[k] += D[k-1]
        if i%2 == 1 : print([D[z] for z in (1..h)])
A071948_triangle(10)  # Peter Luschny, Apr 01 2012

Formula

T(n, n) = A006013(n).

T(n, k) = (n-k+1)binomial(2n+k+1, k)/(n+1) if k<=n.

Let M = the infinite square production matrix

2, 1;

3, 2, 1;

4, 3, 2, 1;

5, 4, 3, 2, 1;

...

The top row of M^n gives reversed terms of n-th row of triangle A071948; with leftmost terms of each row generating A006013 starting (1, 2, 7, 30, 143, ...). - Gary W. Adamson, Jul 07 2011

Extensions

Edited by Emeric Deutsch, Mar 04 2004