A102225 Triangular matrix, read by rows, where row k is formed from the first differences of row (k-1) of its matrix square, with an appended '1' for the main diagonal.

1, 1, 1, 2, -1, 1, 3, -5, 3, 1, 7, -20, 19, -5, 1, 17, -51, 57, -33, 11, 1, 75, -289, 438, -345, 143, -21, 1, 346, -1426, 2441, -2073, 767, -97, 43, 1, 4874, -22622, 44289, -46156, 26231, -7713, 1183, -85, 1, 49047, -259734, 530214, -520395, 272461, -103617, 30735, 1119, 171, 1, 3009094, -15968025, 35495592

Offset

0,4

Comments

Row sums are: {1,2,2,2,2,2,...}. Column 0 is A102226. Column 1 is A102227. Matrix square forms A102228.

Links

Table of n, a(n) for n=0..57.

Formula

T(n, k) = [T^2](n-1, k) - [T^2](n-1, k-1) for n>k>0, with T(n, n)=1 for n>=0 and T(n, 0) = [T^2](n-1, 0) for n>0.

Example

Rows begin:
[1],
[1,1],
[2,-1,1],
[3,-5,3,1],
[7,-20,19,-5,1],
[17,-51,57,-33,11,1],
[75,-289,438,-345,143,-21,1],
[346,-1426,2441,-2073,767,-97,43,1],
[4874,-22622,44289,-46156,26231,-7713,1183,-85,1],...
The matrix square A102225^2 forms A102228:
[1],
[2,1],
[3,-2,1],
[7,-13,6,1],
[17,-34,23,-10,1],
[75,-214,224,-121,22,1],...
The first differences of the rows of A102228
form A102225 excluding the main diagonal of 1's.

Prog

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

Crossrefs

Cf. A102226, A102227, A102228.

Sequence in context: A144866 A058732 A060082 * A183262 A287030 A187066

Adjacent sequences: A102222 A102223 A102224 * A102226 A102227 A102228

Keyword

sign,tabl

Author

Paul D. Hanna, Jan 01 2005

Status

approved