login
Triangle T(n,m) = Sum_{i=0..n-m} C(2*m-1,n-i-1)*C(n-i-1,i)*C(n-i-1,n-m-i)/(2*m-1).
0

%I #19 Oct 07 2020 13:04:30

%S 1,1,1,1,3,2,0,5,10,5,0,5,26,35,14,0,3,44,125,126,42,0,1,52,295,574,

%T 462,132,0,0,44,505,1736,2562,1716,429,0,0,26,655,3864,9450,11220,

%U 6435,1430,0,0,10,655,6664,25830,48840,48477,24310,4862

%N Triangle T(n,m) = Sum_{i=0..n-m} C(2*m-1,n-i-1)*C(n-i-1,i)*C(n-i-1,n-m-i)/(2*m-1).

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

%e The triangle begins

%e 1,

%e 1, 1,

%e 1, 3, 2,

%e 0, 5, 10, 5,

%e 0, 5, 26, 35, 14,

%e 0, 3, 44, 125, 126, 42,

%e 0, 1, 52, 295, 574, 462, 132,

%e 0, 0, 44, 505, 1736, 2562, 1716, 429,

%e 0, 0, 26, 655, 3864, 9450, 11220, 6435, 1430,

%e 0, 0, 10, 655, 6664, 25830, 48840, 48477, 24310, 4862,

%e 0, 0, 2, 505, 9156, 55314, 158136, 243243, 207350, 92378, 16796,

%e 0, 0, 0, 295, 10164, 95844, 403656, 909909, 1178320, 880022, ...

%e From _Peter Luschny_, Oct 07 2020: (Start)

%e Let C(n) denote the Catalan numbers, then the columns start, written as rows,

%e C(0)*[1, 1, 1],

%e C(1)*[1, 3, 5, 5, 3, 1],

%e C(2)*[1, 5, 13, 22, 26, 22, 13, 5, 1],

%e C(3)*[1, 7, 25, 59, 101, 131, 131, 101, 59, 25, 7, 1],

%e C(4)*[1, 9, 41, 124, 276, 476, 654, 726, 654, 476, 276, 124, 41, 9, 1], ... . (End)

%t T[n_, m_] := Sum[Binomial[2*m - 1, n - i - 1] * Binomial[n - i - 1, i] * Binomial[n - i - 1, n - m - i]/(2*m - 1), {i, 0, n - m}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Amiram Eldar_, Oct 07 2020 *)

%o (Maxima)

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

%Y Cf. A000108 (main diagonal), A001700 (1st subdiagonal).

%K nonn,tabl

%O 1,5

%A _Vladimir Kruchinin_, Oct 06 2020