OFFSET
1,5
FORMULA
G.f.: (1-sqrt(1-4*x*(x+1)*(x^2+x+1)*y))/(2*(x+1)).
EXAMPLE
The triangle begins
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, 462, 132,
0, 0, 44, 505, 1736, 2562, 1716, 429,
0, 0, 26, 655, 3864, 9450, 11220, 6435, 1430,
0, 0, 10, 655, 6664, 25830, 48840, 48477, 24310, 4862,
0, 0, 2, 505, 9156, 55314, 158136, 243243, 207350, 92378, 16796,
0, 0, 0, 295, 10164, 95844, 403656, 909909, 1178320, 880022, ...
From Peter Luschny, Oct 07 2020: (Start)
Let C(n) denote the Catalan numbers, then the columns start, written as rows,
C(0)*[1, 1, 1],
C(1)*[1, 3, 5, 5, 3, 1],
C(2)*[1, 5, 13, 22, 26, 22, 13, 5, 1],
C(3)*[1, 7, 25, 59, 101, 131, 131, 101, 59, 25, 7, 1],
C(4)*[1, 9, 41, 124, 276, 476, 654, 726, 654, 476, 276, 124, 41, 9, 1], ... . (End)
MATHEMATICA
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 *)
PROG
(Maxima)
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);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Oct 06 2020
STATUS
approved