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%I #20 Mar 31 2016 21:02:21
%S 1,1,2,4,1,10,4,28,13,1,82,44,6,248,153,27,1,770,536,116,8,2440,1889,
%T 486,46,1,7858,6696,1992,240,10,25644,23849,8042,1180,70,1,84618,
%U 85276,32124,5552,430,12,281844,305933,127287,25306,2430,99,1,946338,1100692
%N Triangle read by rows T(n,k) = the number of Dyck paths of semilength n with k UUDDU's, 0<=k<=[(n-1)/2].
%C Row sums are Catalan numbers A000108.
%H Alois P. Heinz, <a href="/A114848/b114848.txt">Rows n = 0..200, flattened</a>
%H A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924. - From _N. J. A. Sloane_, May 05 2012
%F T(n,k) = Sum((-1)^j * binomial(n-1-(j+k), j+k) * binomial(j + k, k) * A000108(n-2(j+k)), j=0..[(n-1)/2]-k).
%F G.f. G = G(t,z) satisfies G = C(z/(z^2(1-t)+1)), where C(z) is g.f. of Catalan numbers.
%e T(4,1) = 4 because there exist 4 Dyck paths with one occurrence of UUDDU : UDUUDDUD, UUDDUDUD, UUDDUUDD, UUUDDUDD.
%e Triangle begins:
%e : 0 : 1;
%e : 1 : 1;
%e : 2 : 2;
%e : 3 : 4, 1;
%e : 4 : 10, 4;
%e : 5 : 28, 13, 1;
%e : 6 : 82, 44, 6;
%e : 7 : 248, 153, 27, 1;
%e : 8 : 770, 536, 116, 8;
%e : 9 : 2440, 1889, 486, 46, 1;
%e : 10 : 7858, 6696, 1992, 240, 10;
%e : 11 : 25644, 23849, 8042, 1180, 70, 1;
%e : 12 : 84618, 85276, 32124, 5552, 430, 12;
%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p `if`(x=0, 1, expand(b(x-1, y+1, [2, 3, 3, 2, 2][t])
%p *`if`(t=5, z, 1) +b(x-1, y-1, [1, 1, 4, 5, 1][t]))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p seq(T(n), n=0..15); # _Alois P. Heinz_, Jun 10 2014
%t For[n = 1, n <= 20, n++, For[k = 0, k <= Floor[(n - 1)/2], k++, Print[Sum[(-1)^j * Binomial[n - 1 - (j + k), j + k] * Binomial[j + k, k] * Binomial[2(n - 2(j + k)), n - 2(j + k)]/(n - 2(j + k) + 1), {j, 0, Floor[(n - 1)/2] - k}]]]]
%Y Cf. A187256, A243752.
%K nonn,tabf
%O 0,3
%A I. Tasoulas (jtas(AT)unipi.gr), Feb 20 2006
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