OFFSET
1,4
LINKS
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
T(n,m) = A037027(2*n-m-1,n-1)*m/n.
T(n,m) = if n<0 or m<0 or n<m then 0 else if n=m then 1 else if m=0 then 0 else T(n-1,m-1)+T(n,m+1)+T(n,m+2). - Vladimir Kruchinin, Apr 21 2016
Assuming the range n>=0 and 0<=k<=n we have T(n,k) = C(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1) for n>=1 and T(0,0) = 1. - Peter Luschny, Apr 25 2016
EXAMPLE
Triangle starts:
1;
1, 1;
3, 2, 1;
10, 7, 3, 1;
38, 26, 12, 4, 1;
154, 105, 49, 18, 5, 1;
...
MAPLE
T := (n, k) -> `if`(n=0, 1, binomial(2*n-k, n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n], -4)*(k+1)/(n+1)): seq(seq(simplify(T(n, k), k=0..n)), n=0..10); # Peter Luschny, Apr 25 2016
MATHEMATICA
T[n_, m_] := T[n, m] = Which[n <= 0 || m <= 0, 0, n < m, 0, n == m, 1, True, T[n-1, m-1] + T[n, m+1] + T[n, m+2]];
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 03 2019, after Vladimir Kruchinin *)
PROG
(Maxima)
T(n, m):=if n<=0 or m<=0 then 0 else if n<m then 0 else if n=m then 1 else T(n-1, m-1)+T(n, m+1)+T(n, m+2); /* Vladimir Kruchinin, Apr 21 2016 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Mar 21 2011
STATUS
approved