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%I #32 Nov 04 2017 20:33:43
%S 1,3,1,9,5,1,28,20,7,1,90,75,35,9,1,297,275,154,54,11,1,1001,1001,637,
%T 273,77,13,1,3432,3640,2548,1260,440,104,15,1,11934,13260,9996,5508,
%U 2244,663,135,17,1,41990,48450,38760,23256,10659,3705,950,170,19,1
%N Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).
%C T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. _Zoran Sunic_ reference) . - _Benoit Cloitre_, Oct 07 2003
%C Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - _Herbert Kociemba_, May 24 2004
%C Number of standard tableaux of shape (n+k+1, n-k-1). - _Emeric Deutsch_, May 30 2004
%C Riordan array (c(x)^3,xc(x)^2) where c(x) is the g.f. of A000108. Inverse array is A109954. - _Paul Barry_, Jul 06 2005
%H Alois P. Heinz, <a href="/A050155/b050155.txt">Rows n = 1..141, flattened</a>
%H R. K. Guy, Catwalks, Sansteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">J. Integer Seq., Vol. 3 (2000), #00.1.6</a>
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%H A. Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414.
%H J. Riordan, <a href="http://www.jstor.org/stable/2005477">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.
%H _Zoran Sunic_, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1n5">Self-Describing Sequences and the Catalan Family Tree</a>, Electronic Journal of Combinatorics, 10 (2003) #N5.
%F Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).
%F For the column k : expansion of x^(k+1)C^(2k+3) where C = (1-(1-4*x)^(1/2)/(2*x) is the g.f. of Catalan numbers A000108. - _Philippe Deléham_, Feb 03 2004
%F T(n, k) = A039599(n, k+1) = A009766(n+k+1, n-k-1) = A033184(n+k+2, 2k+3) . - _Philippe Deléham_, May 28 2005
%F Sum_{k>= 0} T(m, k)*T(n, k) = A000108(m+n) - A000108(m)*A000108(n). - _Philippe Deléham_, May 28 2005
%F T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - _Paul Barry_, Jul 06 2005
%e 1;
%e 3, 1;
%e 9, 5, 1;
%e 28, 20, 7, 1;
%e 90, 75, 35, 9, 1;
%e 297, 275, 154, 54, 11, 1;
%e . . .
%p T:= (n, k)-> (2*k+3)*binomial(2*n, n-k-1)/(n+k+2):
%p seq(seq(T(n, k), k=0..n-1), n=1..10); # _Alois P. Heinz_, Jan 19 2013
%t T[n_, k_] := (2*k + 3)*Binomial[2*n, n - k - 1]/(n + k + 2);
%t Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, May 21 2016 *)
%Y First columns: A000245, A000344, A000588, A001392, A000589, A000590.
%Y Cf. A000108, A001791 (row sums), A050144.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_
%E Edited by _Philippe Deléham_, May 22 2005
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