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

%I A071948 #53 Jan 05 2025 19:51:37
%S A071948 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,
%T A071948 1020,2448,3876,1,14,102,510,1938,5814,13566,21318,1,16,133,760,3325,
%U A071948 11704,33649,76912,120175,1,18,168,1078,5313,21252,70840,197340,444015
%N A071948 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
%C A071948 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
%C A071948 ............1 a_2 ... a_(n-1) a_n..........
%C A071948 ............1 b_2 ... b_(n-1) b_n..........
%C A071948 such that a_i <= i (2 <= i <= n) and b_2 <= a_2 <= ... <= b_n <= a_n = k.
%C A071948 See A193091 and A211788 for other two-line array enumerations. - _Peter Bala_, Aug 02 2012
%H A071948 Michael De Vlieger, <a href="/A071948/b071948.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A071948 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A071948 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
%H A071948 S. Dulucq, <a href="/A005819/a005819.pdf">Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots</a>, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
%H A071948 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%H A071948 D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays</a>, Canad J. Math., 49 (1997), 301-320.
%H A071948 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F A071948 T(n, n) = A006013(n).
%F A071948 T(n, k) = (n-k+1)binomial(2n+k+1, k)/(n+1) if k<=n.
%F A071948 Let M = the infinite square production matrix
%F A071948   2, 1;
%F A071948   3, 2, 1;
%F A071948   4, 3, 2, 1;
%F A071948   5, 4, 3, 2, 1;
%F A071948   ...
%F A071948 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
%e A071948 Triangle begins
%e A071948   1;
%e A071948   1, 2;
%e A071948   1, 4,  7;
%e A071948   1, 6, 18, 30;
%e A071948   1, 8, 33, 88, 143;
%p A071948 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);
%t A071948 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 *)
%o A071948 (Sage) # Computes the first n rows of the triangle.
%o A071948 def A071948_triangle(n) :
%o A071948     D = [0 for i in (0..n+1)]; D[1] = 1
%o A071948     for i in (4..2*n+3) :
%o A071948         h = i//2 - 1
%o A071948         for k in (1..h) : D[k] += D[k-1]
%o A071948         if i%2 == 1 : print([D[z] for z in (1..h)])
%o A071948 A071948_triangle(10)  # _Peter Luschny_, Apr 01 2012
%Y A071948 Row sums give A001764.
%Y A071948 Rows are the reversals of the rows of A092276.
%Y A071948 Cf. A193091, A211788.
%K A071948 nonn,easy,tabl
%O A071948 0,3
%A A071948 _N. J. A. Sloane_, Jun 15 2002
%E A071948 Edited by _Emeric Deutsch_, Mar 04 2004