A111975 Triangle P, read by rows, that satisfies [P^2](n,k) = P(n+1,k+1) for n>=k>=0, also [P^(2*m)](n,k) = [P^m](n+1,k+1) for all m, where [P^m](n,k) denotes the element at row n, column k, of the matrix power m of P, with P(k,k)=1 and P(k+2,2)=P(k+2,0) for k>=0.

1, 1, 1, 1, 2, 1, 4, 4, 4, 1, 16, 16, 16, 8, 1, 96, 96, 96, 64, 16, 1, 896, 896, 896, 704, 256, 32, 1, 13568, 13568, 13568, 11776, 5504, 1024, 64, 1, 345088, 345088, 345088, 317952, 178176, 43776, 4096, 128, 1, 15112192, 15112192, 15112192, 14422016

Offset

0,5

Comments

Terms of column 0, column 1 and column 2 in row n are equal for n>2.

Links

Table of n, a(n) for n=0..48.

Formula

The g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*2^k)^n/n! * Product_{j=0..n-1} L(2^j*x) where L(x) is the g.f. of column 0 of the matrix log of P (A111979) and satisfies: x-x^2 = Sum_{j>=1}(1-2^j*x)*Prod_{i=0..j-1}L(2^i*x).

Example

Triangle P begins:
1;
1,1;
1,2,1;
4,4,4,1;
16,16,16,8,1;
96,96,96,64,16,1;
896,896,896,704,256,32,1;
13568,13568,13568,11776,5504,1024,64,1;
345088,345088,345088,317952,178176,43776,4096,128,1; ...
where P^2 shifts columns left and up one place:
1;
2,1;
4,4,1;
16,16,8,1;
96,96,64,16,1; ...
The matrix inverse, P^-1, equals signed P:
1;
-1,1;
1,-2,1;
-4,4,-4,1;
16,-16,16,-8,1; ...

Prog

(PARI) P(n, k, q=2)=local(A=Mat(1), B); if(n<k || k<0, 0, for(m=1, 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, j]=if(i>2, (A^q)[i-1, 2], 1), B[i, j]=(A^q)[i-1, j-1])); )); A=B); return(A[n+1, k+1]))

Crossrefs

Cf. A111976 (column 0), A111977 (row sums), A111978 (matrix log), A098539 (variant), A078536 (variant).

Sequence in context: A008741 A369999 A110316 * A117250 A345674 A296337

Adjacent sequences: A111972 A111973 A111974 * A111976 A111977 A111978

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 24 2005

Status

approved