OFFSET
0,9
LINKS
Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
FORMULA
T(n,m) = C(n-1,n-m)*Sum_{k=0..n} C(n-m,m-k)*k/m, T(0,0)=1, T(0,m)=0, m>0.
Sum_{m=0..n} T(n,m) = A005773(n).
m*T(m, n) + (-2*m + 6)*T(m - 3, n) + (4*m - 12)*T(m - 3, n - 2) + (7*m - 21)*T(m - 3, n - 1) + (5*m - 9)*T(m - 2, n) + (-2*m + 9)*T(m - 2, n - 1) + (-4*m + 3)*T(m - 1, n) - m*T(m - 1, n - 1)= 0 for m>=3, n>=2. - Robert Israel, Oct 05 2020
EXAMPLE
1,
0,1,
0,1, 1,
0,1, 3, 1,
0,1, 6, 5, 1,
0,1,10,16, 7, 1,
0,1,15,40, 30, 9, 1,
0,1,21,85,100,48,11,1
MAPLE
T:= proc(m, n) option remember;
if m < n then 0
elif m = n then 1
elif n=0 then 0
elif n=1 then 1
else
(-(-2*m + 6)*procname(m - 3, n) - (4*m - 12)*procname(m - 3, n - 2) - (7*m - 21)*procname(m - 3, n - 1) - (5*m - 9)*procname(m - 2, n) - (-2*m + 9)*procname(m - 2, n - 1) - (-4*m + 3)*procname(m - 1, n) + m*procname(m - 1, n - 1))/m
fi
end proc:
seq(seq(T(n, m), m=0..n), n=0..20); # Robert Israel, Oct 05 2020
MATHEMATICA
T[n_, m_] := If[n == m, 1, If[m == 0, 0, Binomial[n-1, n-m]*
Sum[Binomial[n-m, m-k]*k, {k, 0, n}]/m]];
Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)
PROG
(Maxima)
T(n, m):=if m=0 and n=0 then 1 else if m=0 then 0 else (binomial(n-1, n-m)*sum(binomial(n-m, m-k)*k, k, 0, n))/m;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Oct 04 2020
STATUS
approved