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%I #23 Mar 05 2017 01:39:42
%S 1,2,1,4,4,1,12,12,6,1,40,40,24,8,1,144,144,92,40,10,1,544,544,360,
%T 176,60,12,1,2128,2128,1440,752,300,84,14,1,8544,8544,5872,3200,1400,
%U 472,112,16,1,35008,35008,24336,13664,6352,2400,700,144,18,1
%N Triangle T(n,k) = coefficient of x^n in expansion of ((1 -sqrt(1 - 4*x - 4*x^2))/2)^k.
%C Triangle T(n,k) =
%C 1. Riordan Array (1, (1 - sqrt(1 - 4*x - 4*x^2))/2) without first column.
%C 2. Riordan Array ((1 - sqrt(1 - 4*x - 4*x^2))/(2*x), (1 - sqrt(1 - 4*x - 4*x^2))/2) numbering triangle (0,0).
%C The array factorizes in the Bell subgroup of the Riordan group as (1 + x, x*(1 + x)) * (c(x), x*c(x)) = A030528 * A033184, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - _Peter Bala_, Dec 11 2015
%F T(n,k) = k*( Sum_{i = k..n} binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i );
%e 1,
%e 2, 1,
%e 4, 4, 1,
%e 12, 12, 6, 1,
%e 40, 40, 24, 8, 1,
%e 144, 144, 92, 40, 10, 1,
%e 544, 544, 360, 176, 60, 12, 1
%t Table[k Sum[Binomial[i, n - i] Binomial[-k + 2 i - 1, i - 1]/i, {i, k, n}], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Dec 11 2015 *)
%o (Maxima) T(n,k):=k*(sum((binomial(i,n-i)*binomial(-k+2*i-1,i-1))/i,i,k,n));
%o (PARI)
%o tabl(nn) = {for (n=1, nn, for(k=1, n, print1(k*sum(i=k, n, binomial(i,n-i)*binomial(-k+2*i-1,i-1)/i),", ",);); print();); };
%o tabl(10); \\ _Indranil Ghosh_, Mar 04 2017
%Y Cf. A025227 (column 1), A000108, A030528, A033184.
%K nonn,tabl
%O 1,2
%A _Vladimir Kruchinin_, Nov 22 2011
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