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%I #14 Oct 06 2020 02:45:11
%S 1,1,1,1,3,2,1,6,8,5,1,10,22,26,14,1,15,50,85,90,42,1,21,100,225,348,
%T 322,132,1,28,182,525,1050,1442,1176,429,1,36,308,1120,2730,4928,5992,
%U 4356,1430,1,45,492,2226,6426,14238,22920,24894,16302,4862
%N Triangle T(n,m) = C(n-1,n-m)*Sum_{k=1..n} C(2*k-2,k-1)*C(n-m,m-k)/m, m>0, n>0, n>=m.
%F G.f.: A(x,y) = -(sqrt((2*sqrt(-4*x^2*y+x^2-2*x+1)+3*x-2)/(4*x))-1/2).
%e 1,
%e 1, 1,
%e 1, 3, 2,
%e 1, 6, 8, 5,
%e 1,10, 22, 26, 14,
%e 1,15, 50, 85, 90, 42,
%e 1,21,100,225,348,322,132
%t Table[Binomial[n - 1, n - m] Sum[Binomial[2 k - 2, k - 1] Binomial[n - m, m - k]/m, {k, n}], {n, 10}, {m, n}] // Flatten (* _Michael De Vlieger_, Oct 05 2020 *)
%o (Maxima)
%o T(n,m):=(binomial(n-1,n-m)*sum(binomial(2*k-2,k-1)*binomial(n-m,m-k),k,1,n))/m;
%Y T(2*n,n) is A069720.
%Y 2nd column: A000217, 3rd column: 2*A006522 or 2*(A027927-1).
%K nonn,tabl
%O 1,5
%A _Vladimir Kruchinin_, Oct 05 2020
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