A096744 - OEIS

%I #10 Nov 30 2016 12:32:28

%S 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,

%T 138,55,12,3,1,1,1,429,498,219,55,12,3,1,1,1,1696,2103,969,273,55,12,

%U 3,1,1,1,6583,8734,4215,1266,273,55,12,3,1,1,1,29167,40291,19989,6213,1428

%N Triangle T, read by rows, such that the matrix cube shifts T one place diagonally left and upward, with T(n, 0) = T(n, n) = 1 for n>=0.

%C Column with index 1 forms A096745 shift right. Row sums form A096746. The convergent of the rows in reverse order is the sequence enumerating ternary trees (A001764).

%F 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.

%e Rows begin:

%e {1},

%e {1,1},

%e {1,1,1},

%e {1,3,1,1},

%e {1,6,3,1,1},

%e {1,16,12,3,1,1},

%e {1,40,37,12,3,1,1},

%e {1,133,138,55,12,3,1,1},

%e {1,429,498,219,55,12,3,1,1},

%e {1,1696,2103,969,273,55,12,3,1,1},

%e {1,6583,8734,4215,1266,273,55,12,3,1,1},

%e {1,29167,40291,19989,6213,1428,273,55,12,3,1,1},

%e {1,128401,185364,94755,30249,7266,1428,273,55,12,3,1,1},...

%e T^3 begins:

%e {1},

%e {3,1},

%e {6,3,1},

%e {16,12,3,1},

%e {40,37,12,3,1},

%e {133,138,55,12,3,1},

%e {429,498,219,55,12,3,1},...

%e which equals T shift one place diagonally left and upward.

%t 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 *)

%o (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) )))}

%Y Cf. A096745, A096746, A096591, A001764.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Jul 06 2004