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

%I A030237 #49 Mar 27 2022 11:13:37
%S A030237 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 A030237 429,1,8,35,110,275,572,1001,1430,1,9,44,154,429,1001,2002,3432,4862,
%U A030237 1,10,54,208,637,1638,3640,7072,11934,16796,1,11,65,273,910,2548,6188,13260,25194,41990,58786
%N A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).
%C A030237 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
%H A030237 Reinhard Zumkeller, <a href="/A030237/b030237.txt">Rows n=1..151 of triangle, flattened</a>
%H A030237 B. Derrida, E. Domany and D. Mukamel, <a href="http://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%H A030237 Wolfdieter Lang, <a href="/A030237/a030237.txt">First 10 rows.</a>
%H A030237 Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 6.
%F A030237 T(n, k) = (n-k+1)*binomial(n+k, k)/(n+1).
%F A030237 Sum_{k=0..n-1} T(n,k) = A000245(n). - _G. C. Greubel_, Mar 17 2021
%F A030237 T(n, k) = [x^k] ((1 - 2*x)/(1 - x)^(n + 2)). - _Peter Luschny_, Mar 27 2022
%e A030237 Triangle begins as:
%e A030237   1;
%e A030237   1, 2;
%e A030237   1, 3,  5;
%e A030237   1, 4,  9,  14;
%e A030237   1, 5, 14,  28,  42;
%e A030237   1, 6, 20,  48,  90,  132;
%e A030237   1, 7, 27,  75, 165,  297,  429;
%e A030237   1, 8, 35, 110, 275,  572, 1001, 1430;
%e A030237   1, 9, 44, 154, 429, 1001, 2002, 3432, 4862;
%p A030237 A030237 := proc(n,m)
%p A030237     (n-m+1)*binomial(n+m,m)/(n+1) ;
%p A030237 end proc: # _R. J. Mathar_, May 31 2016
%p A030237 # Compare the analogue algorithm for the Bell numbers in A011971.
%p A030237 CatalanTriangle := proc(len) local P, T, n; P := [1]; T := [[1]];
%p A030237 for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), P[-1]]);
%p A030237 T := [op(T), P] od; T end: CatalanTriangle(6):
%p A030237 ListTools:-Flatten(%); # _Peter Luschny_, Mar 26 2022
%p A030237 # Alternative:
%p A030237 ogf := n -> (1 - 2*x)/(1 - x)^(n + 2):
%p A030237 ser := n -> series(ogf(n), x, n):
%p A030237 row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
%p A030237 seq(row(n), n = 1..11); # _Peter Luschny_, Mar 27 2022
%t A030237 T[n_, k_]:= T[n, k] = Which[k==0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1]];
%t A030237 Table[T[n, k], {n,1,12}, {k,0,n-1}] // Flatten (* _Jean-François Alcover_, Nov 14 2017 *)
%o A030237 (Haskell)
%o A030237 a030237 n k = a030237_tabl !! n !! k
%o A030237 a030237_row n = a030237_tabl !! n
%o A030237 a030237_tabl = map init $ tail a009766_tabl
%o A030237 -- _Reinhard Zumkeller_, Jul 12 2012
%o A030237 (PARI) T(n,k) = (n-k+1)*binomial(n+k, k)/(n+1) \\ _Andrew Howroyd_, Feb 23 2018
%o A030237 (Sage) flatten([[(n-k+1)*binomial(n+k, k)/(n+1) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Mar 17 2021
%o A030237 (Magma) [(n-k+1)*Binomial(n+k, k)/(n+1): k in [0..n-1], n in [1..12]]; // _G. C. Greubel_, Mar 17 2021
%Y A030237 Alternate versions of (essentially) the same Catalan triangle: A009766, A033184, A047072, A059365, A099039, A106566, A130020.
%Y A030237 Diagonals give A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003518, A003519.
%Y A030237 Row sums give A071724.
%Y A030237 Cf. A009766, A350584.
%K A030237 nonn,tabl,easy
%O A030237 1,3
%A A030237 _Wouter Meeussen_
%E A030237 Missing a(8) = T(7,0) = 1 inserted by _Reinhard Zumkeller_, Jul 12 2012