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A113206 Triangle read by rows of generalized Catalan numbers.

%I A113206 #16 Nov 28 2017 10:10:09
%S A113206 1,1,0,1,1,0,2,0,1,1,0,3,5,3,0,1,1,0,4,12,14,12,4,0,1,1,0,5,22,55,42,
%T A113206 55,22,5,0,1,1,0,6,35,140,273,132,273,140,35,6,0,1,1,0,7,51,285,969,
%U A113206 1428,429,1428,969,285,51,7,0,1,1,0,8,70,506,2530,7084,7752,1430,7752,7084
%N A113206 Triangle read by rows of generalized Catalan numbers.
%C A113206 A dual to Pascal's triangle. Row n has 2n+1 entries.
%H A113206 Ph. Leroux, <a href="http://arxiv.org/abs/math.CO/0512437">A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers</a>, arXiv:math/0512437 [math.CO], 2005.
%H A113206 Ph. Leroux, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Leroux/leroux1.html">A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers</a>, J. Integer Seqs., 10 (2007), #07.4.7.
%F A113206 T(n,k) = A070914(n-|n-k|-1,|n-k|+1) if 3<=k<=2n-3 . - _R. J. Mathar_, Feb 08 2008
%e A113206 .............1
%e A113206 ...........1.0.1
%e A113206 .........1.0.2.0.1
%e A113206 .......1.0.3.5.3.0.1
%e A113206 ....1.0.4.12.14.12.4.0.1
%e A113206 .1.0.5.22.55.42.55.22.5.0.1
%p A113206 A070914 := proc(n,k) binomial(n*(k+1),n)/(n*k+1) ; end proc:
%p A113206 A113206 := proc(n,k) if k = 2 or k = 2*n-2 then 0 ; else A070914(n-abs(n-k)-1,abs(n-k)+1) ; fi ; end proc:
%p A113206 for n from 0 to 10 do for k from 1 to 2*n-1 do printf("%d ",A113206(n,k)) ; od: od: # _R. J. Mathar_, Feb 08 2008
%t A113206 A070914[n_, k_] := Binomial[n*(k + 1), n]/(n*k + 1);
%t A113206 A113206[n_, k_] := If[k == 2 || k == 2*n - 2, 0, A070914[n - Abs[n-k] - 1, Abs[n-k] + 1]];
%t A113206 Table[A113206[n, k], {n, 0, 10}, {k, 1, 2*n - 1}] // Flatten (* _Jean-François Alcover_, Nov 28 2017, after _R. J. Mathar_ *)
%Y A113206 Cf. A007318, A000108.
%K A113206 nonn,tabf,easy
%O A113206 0,7
%A A113206 _N. J. A. Sloane_, Jan 07 2006