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%I #12 Mar 04 2017 13:52:19
%S 1,1,1,3,2,1,6,7,3,1,16,18,12,4,1,47,53,37,18,5,1,146,162,120,64,25,6,
%T 1,471,518,390,227,100,33,7,1,1562,1708,1299,788,385,146,42,8,1,5291,
%U 5762,4410,2750,1426,606,203,52,9,1,18226,19788,15216,9664,5225,2388,903,272,63,10,1
%N Triangle, read by rows, T(n,k) = k*Sum_{i=0..(n-k)/2} C(k,i)*C(2*n-k-4*i-1,n-2*i-k)/(n-2*i).
%H Indranil Ghosh, <a href="/A257558/b257558.txt">Rows 1..100, flattened</a>
%F G.f.: 1/(1-C(x)*(x+x^3)*y)-1, where C(x) is g.f. of Catalan numbers (A000108).
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 6, 7, 3, 1;
%e 16, 18, 12, 4, 1;
%e 47, 53, 37, 18, 5, 1;
%t Flatten[Table[k*Sum[Binomial[k,i] * Binomial[2n-k-4i-1,n-2i-k] / (n-2i), {i, 0, (n-k)/2}], {n, 1, 11},{k, 1, n}]] (* _Indranil Ghosh_, Mar 04 2017 *)
%o (Maxima)
%o T(n,k):=(k*sum((binomial(k,i)*binomial(2*n-k-4*i-1,n-2*i-k))/(n-2*i),i,0,(n-k)/2));
%o (PARI)
%o tabl(nn) = {for (n=1, nn, for(k=1, n, print1(k*sum(i=0,(n-k)/2, binomial(k,i) * binomial(2*n-k-4*i-1,n-2*i-k) / (n-2*i)),", ");); print(););};
%o tabl(11); \\ _Indranil Ghosh_, Mar 04 2017
%Y Cf. A000108.
%K nonn,tabl
%O 1,4
%A _Vladimir Kruchinin_, Apr 30 2015
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