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%I #43 Jun 23 2018 04:56:41
%S 1,1,2,1,5,5,14,20,1,42,75,9,132,275,54,1,429,1001,273,13,1430,3640,
%T 1260,104,1,4862,13260,5508,663,17,16796,48450,23256,3705,170,1,58786,
%U 177650,95931,19019,1309,21,208012,653752,389367,92092,8602,252,1
%N Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.
%C Closely related to triangle A118919.
%C Row n contains 1+floor(n/2) terms.
%C From _Peter Bala_, Mar 20 2009: (Start)
%C Combinatorial interpretations of T(n,k):
%C 1) The number of standard tableaux of shape (n-2*k,n+2*k).
%C 2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)
%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>, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009
%F G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
%F Row sums equal A088218(n) = C(2*n-1,n).
%F T(n,0) = A000108(n) (the Catalan numbers).
%F T(n,1) = A000344(n).
%F T(n,2) = A001392(n).
%F Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
%F Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
%F ...
%F T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - _Peter Bala_, Mar 20 2009
%F T(n,k) = A039599(n, 2*k). - _Johannes W. Meijer_, Sep 04 2013
%F A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - _Bradley Klee_, Feb 26 2018
%e Triangle begins:
%e 1;
%e 1;
%e 2, 1;
%e 5, 5;
%e 14, 20, 1;
%e 42, 75, 9;
%e 132, 275, 54, 1;
%e 429, 1001, 273, 13;
%e 1430, 3640, 1260, 104, 1;
%e 4862, 13260, 5508, 663, 17; ...
%t f1 = (1-Sqrt[1-4*x])/(2*x);
%t DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm (* _Bradley Klee_, Feb 26 2018 *)
%t Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* _Bradley Klee_, Feb 26 2018 *)
%o (PARI) T(n,k)=(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)
%Y Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.
%K nonn,tabf
%O 0,3
%A _Paul D. Hanna_, May 10 2006
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