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A071948 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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.

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

LINKS

Table of n, a(n) for n=0..53.

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'Universite 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

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 = []; [D.append(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

Row sums give A001764.

Rows are the reversals of the rows of A092276.

Cf. A193091, A211788.

Sequence in context: A052566 A234946 A223092 * A193589 A187115 A121722

Adjacent sequences:  A071945 A071946 A071947 * A071949 A071950 A071951

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane, Jun 15 2002

EXTENSIONS

Edited by Emeric Deutsch, Mar 04 2004

STATUS

approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)