A113983 Triangle, read by rows, such that T(n,k) = T(n-1,k-1) + [T^2](n-2,k-1) with T(n,0) = T(n,n) = 1 for n>=0, k>=0.

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

Offset

0,5

Comments

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).

Links

Table of n, a(n) for n=0..65.

Formula

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.

Example

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.

Prog

(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]

Crossrefs

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

Sequence in context: A144048 A292193 A258708 * A199333 A089980 A181031

Adjacent sequences: A113980 A113981 A113982 * A113984 A113985 A113986

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 12 2005

Status

approved