OFFSET
0,3
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.
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.
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
EXAMPLE
Triangle begins
1;
1, 2;
1, 4, 7;
1, 6, 18, 30;
1, 8, 33, 88, 143;
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Jun 15 2002
EXTENSIONS
Edited by Emeric Deutsch, Mar 04 2004
STATUS
approved