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Triangle, read by rows, T(n,k) = (k+1)*Sum_{i=0..n-k} C(k+2*i,i)*C(n-i-1,n-k-i)/(k+i+1).
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%I #52 Apr 22 2015 16:22:29

%S 1,1,1,2,3,1,5,8,5,1,14,22,18,7,1,42,64,59,32,9,1,132,196,190,124,50,

%T 11,1,429,625,618,448,225,72,13,1,1430,2055,2047,1576,910,370,98,15,1,

%U 4862,6917,6908,5510,3482,1664,567,128,17,1,16796,23713,23703,19322,12958,6924,2814,824,162,19,1

%N Triangle, read by rows, T(n,k) = (k+1)*Sum_{i=0..n-k} C(k+2*i,i)*C(n-i-1,n-k-i)/(k+i+1).

%F G.f.: 1/(1-(1-sqrt(1-4*x))/2-y*(x/(1-x))).

%e 1;

%e 1, 1;

%e 2, 3, 1;

%e 5, 8, 5, 1;

%e 14, 22, 18, 7, 1;

%e 42, 64, 59, 32, 9, 1;

%o (Maxima) T(n,k):=(k+1)*sum((binomial(k+2*i,i)*binomial(n-i-1,n-k-i))/(k+i+1),i,0,n-k);

%Y Cf. A000108 (first column), A014138.

%K nonn,tabl

%O 0,4

%A _Vladimir Kruchinin_, Apr 21 2015