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A338022
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
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
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
Cf. A000108 (main diagonal), A001700 (1st subdiagonal).
Sequence in context: A246773 A359843 A364361 * A253176 A079408 A114376
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Oct 06 2020
STATUS
approved