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A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n). 23

%I

%S 1,1,2,1,3,5,1,4,9,14,1,5,14,28,42,1,6,20,48,90,132,1,7,27,75,165,297,

%T 429,1,8,35,110,275,572,1001,1430,1,9,44,154,429,1001,2002,3432,4862

%N Catalan's triangle with right border removed (n > 0, 0 <= k < n).

%C This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - _Wolfdieter Lang_, Jan 13 2006.

%D B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

%H Reinhard Zumkeller, <a href="/A030237/b030237.txt">Rows n=1..151 of triangle, flattened</a>

%H W. Lang: <a href="http://www.itp.kit.edu/~wl/EISpub/A030237.text">First 10 rows.</a>

%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757, 2015. See Fig. 6.

%F T(n,m) = (n-m+1)*binomial(n+m,m)/(n+1).

%e 1;

%e 1,2;

%e 1,3,5;

%e 1,4,9,14;

%e 1,5,14,28,42;

%e 1,6,20,48,90,132;

%e 1,7,27,75,165,297,429;

%e 1,8,35,110,275,572,1001,1430;

%e 1,9,44,154,429,1001,2002,3432,4862;

%p A030237 := proc(n,m)

%p (n-m+1)*binomial(n+m,m)/(n+1) ;

%p end proc: # _R. J. Mathar_, May 31 2016

%t T[n_, k_] := T[n, k] = Which[k==0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1]];

%t Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* _Jean-Fran├žois Alcover_, Nov 14 2017 *)

%o (Haskell)

%o a030237 n k = a030237_tabl !! n !! k

%o a030237_row n = a030237_tabl !! n

%o a030237_tabl = map init $ tail a009766_tabl

%o -- _Reinhard Zumkeller_, Jul 12 2012

%o (PARI) T(n,k) = (n-k+1)*binomial(n+k, k)/(n+1) \\ _Andrew Howroyd_, Feb 23 2018

%Y Cf. A009766.

%Y Row sums give A071724(n).

%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

%Y Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

%K nonn,tabl,easy,changed

%O 1,3

%A _Wouter Meeussen_

%E Missing a(8) = T(7,0) = 1 inserted by _Reinhard Zumkeller_, Jul 12 2012

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Last modified February 23 22:59 EST 2018. Contains 299595 sequences. (Running on oeis4.)