A098539 Lower triangular matrix T, read by rows, that shifts left one column under the matrix square of T, with T(n,0)=T(n,1) for n>0 and T(n,n)=1 for n>=0.

1, 1, 1, 2, 2, 1, 6, 6, 4, 1, 26, 26, 20, 8, 1, 166, 166, 140, 72, 16, 1, 1626, 1626, 1460, 888, 272, 32, 1, 25510, 25510, 23884, 16392, 6256, 1056, 64, 1, 664666, 664666, 639156, 479736, 215696, 46816, 4160, 128, 1, 29559718, 29559718, 28895052

Offset

0,4

Comments

Column 0 forms A002449, the number of different types of binary trees of height n. Row sums form A098540. Column 1 equals A098541. As a matrix, T satisfies [T^2](n,k) = T(n+1,k+1) for all n,k>=0, where [T^2] denotes the matrix square of T, with T(0,k)=[T^2](k,0) and T(k,k)=1 for all k>=0. This is a variant of triangle A078121.

Links

Table of n, a(n) for n=0..47.

Formula

T(n, 0) = A002449(n), T(n, n)=1 for n>=0; T(n, 1)=T(n, 0) for n>0.

Example

Rows of T begin:
[1],
[1,1],
[2,2,1],
[6,6,4,1],
[26,26,20,8,1],
[166,166,140,72,16,1],
[1626,1626,1460,888,272,32,1],
[25510,25510,23884,16392,6256,1056,64,1],
[664666,664666,639156,479736,215696,46816,4160,128,1],...
Matrix square T^2 begins:
[1],
[2,1],
[6,4,1],
[26,20,8,1],
[166,140,72,16,1],
[1626,1460,888,272,32,1],...
showing that T^2 is the same as T after shifting left one column.

Prog

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

Crossrefs

Cf. A002449, A098540, A078121.

Sequence in context: A108076 A104557 A141712 * A222073 A343847 A135880

Adjacent sequences: A098536 A098537 A098538 * A098540 A098541 A098542

Keyword

nonn,tabl

Author

Paul D. Hanna, Sep 13 2004

Status

approved