OFFSET
0,8
COMMENTS
FORMULA
T(n, k) = Sum_{i=0..n-1} Sum_{j=0..n-2} T(n-2, j)*T(j, i)*T(i, k-1) for n>1 and k>0; T(n, 0)=T(n, n)=1.
EXAMPLE
Rows begin:
{1},
{1,1},
{1,1,1},
{1,3,1,1},
{1,6,3,1,1},
{1,16,12,3,1,1},
{1,40,37,12,3,1,1},
{1,133,138,55,12,3,1,1},
{1,429,498,219,55,12,3,1,1},
{1,1696,2103,969,273,55,12,3,1,1},
{1,6583,8734,4215,1266,273,55,12,3,1,1},
{1,29167,40291,19989,6213,1428,273,55,12,3,1,1},
{1,128401,185364,94755,30249,7266,1428,273,55,12,3,1,1},...
T^3 begins:
{1},
{3,1},
{6,3,1},
{16,12,3,1},
{40,37,12,3,1},
{133,138,55,12,3,1},
{429,498,219,55,12,3,1},...
which equals T shift one place diagonally left and upward.
MATHEMATICA
T[n_, 0] := 1; T[n_, n_] := 1; T[n_, k_] := Sum[ Sum[ T[n - 2, j]*T[j, i]*T[i, k - 1], {j, 0, n - 2}], {i, 0, n - 1}] ; Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 28 2016 *)
PROG
(PARI) {T(n, k) = if(n<k||k<0, 0 , if(n==k||k==0, 1, sum(i=0, n-1, sum(j=0, n-2, T(n-2, j)*T(j, i))*T(i, k-1) )))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 06 2004
STATUS
approved