0,5

Surprisingly, T(n,k) = [T^k](n-k,0) + [T^(k+1)](n-k-1,0), where T^k is the k-th power of T as a triangular matrix. See triangle A113993, where column k of A113993 equals column 0 of T^(k+1).

T(n, k) = [T^k](n-k, 0) + [T^(k+1)](n-k-1, 0), or, equivalently, T(n, k) = A113993(n-1, k-1) + A113993(n-1, k). From the definition, G.f. satisfies: A(x, y) = 1/(1-x) + x*y*A(x, y) + x^2*y*GF(T^2), where GF(T^2) equals the g.f. of the matrix square of T.

Triangle T begins:

1;

1,1;

1,2,1;

1,3,3,1;

1,5,7,4,1;

1,9,17,13,5,1;

1,19,45,43,21,6,1;

1,47,135,153,89,31,7,1;

1,137,463,603,401,161,43,8,1;

1,465,1817,2657,1969,881,265,57,9,1;

1,1819,8121,13111,10633,5191,1709,407,73,10,1;

1,8123,41075,72273,63297,33223,11759,3025,593,91,11,1; ...

Matrix square, T^2 (=A113987), begins:

1;

2,1;

4,4,1;

8,12,6,1;

18,36,26,8,1;

46,116,108,46,10,1;

136,416,468,248,72,12,1; ...

where T(n,k) = T(n-1,k-1) + [T^2](n-2,k-1):

T(8,2) = 463 = T(7,1) + [T^2](6,1) = 47 + 416;

T(8,3) = 603 = T(7,2) + [T^2](6,2) = 135 + 468;

T(8,4) = 401 = T(7,3) + [T^2](6,3) = 153 + 248.

(PARI) {T(n, k)=local(A, B); A=Mat(1); for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(i<3|j==1|j==i, B[i, j]=1, B[i, j]=A[i-1, j-1]+(A^2)[i-2, j-1]); )); A=B); A[n+1, k+1]}

Cf. A113984 (column 1), A113985 (column 2), A113986 (column 3), A113987 (column 4); A113988 (T^2), A113993.

Sequence in context: A138201 A026736 A050446 this_sequence A089980 A128545 A034364

Adjacent sequences: A113980 A113981 A113982 this_sequence A113984 A113985 A113986

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 12 2005