OFFSET
0,5
FORMULA
T(n, k) = a(n, k+1) * a(n, k+2) * a(n, k+3)/(n+1), where a(n, k) is the solution to a(n, k) = a(n, k-1) + n*a(n, k-2) for k >= 2 with a(n, 0) = 0 and a(n, 1) = 1 for all n >= 0.
Row n has g.f. 1/((1 + n*x - n^3*x^2) * (1 - (3*n + 1)*x - n^3*x^2)).
EXAMPLE
Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 3, 15, 60, 260, 1092, 4641, 19635, ...
1, 5, 55, 385, 3311, 25585, 208335, 1652145, ...
1, 7, 133, 1330, 18430, 210490, 2673223, 31940881, ...
1, 9, 261, 3393, 68237, 1037673, 18598293, 300963537, ...
1, 11, 451, 7216, 197456, 3761296, 89565861, 1842200151, ...
...
MAPLE
a := proc(n, k) local v; option remember; if k = 0 and 0 <= n then v := 0; end if; if k = 1 and 0 <= n then v := 1; end if; if 2 <= k and 0 <= n then v := a(n, k - 1) + n*a(n, k - 2); end if; v; end proc;
T := proc(n, k) a(n, k + 1)*a(n, k + 2)*a(n, k + 3)/(n + 1); end proc;
seq(seq(T(k, n-k), k=0..n), n=0..10); # Petros Hadjicostas, Dec 26 2019
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 12 2005
STATUS
approved