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A337991
Triangle T(n,m)= Sum_{i=1..n} C(n,i-m)*C(n+m-i,i-1)*C(n+m-i,m)/n, T(0,0)=1.
0
1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 4, 13, 15, 7, 1, 9, 35, 52, 36, 11, 1, 21, 96, 175, 160, 75, 16, 1, 51, 267, 576, 655, 415, 141, 22, 1, 127, 750, 1869, 2541, 2030, 952, 245, 29, 1, 323, 2123, 6000, 9492, 9156, 5488, 1988, 400, 37, 1, 835, 6046, 19107, 34476, 38976, 28476, 13356, 3852, 621, 46, 1
OFFSET
0,5
FORMULA
G.f.: (-sqrt(x^2*(y^2-2*y-3)+x*(-2*y-2)+1)-x*(y-1)+1)/(2*x).
EXAMPLE
1,
1, 1,
1, 2, 1,
2, 5, 4, 1,
4, 13, 15, 7, 1,
9, 35, 52, 36, 11, 1,
21, 96, 175, 160, 75, 16, 1,
51, 267, 576, 655, 415, 141, 22, 1
MATHEMATICA
T[0, 0] = 1; T[n_, m_] := Sum[Binomial[n, i - m] * Binomial[n + m - i, i - 1] * Binomial[n + m - i, m]/n, {i, 1, n}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Amiram Eldar, Oct 06 2020 *)
PROG
(Maxima)
T(n, m):=if m=n then 1 else if n=0 then 0 else sum(binomial(n, i-m)*binomial(n+m-i, i-1)*binomial(n+m-i, m), i, 1, n)/n;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Oct 06 2020
STATUS
approved