0,5

Also P(n,k) = partitions of (8^n - 8^(n-k)) into powers of 8 <= 8^(n-k).

Let q=8; the g.f. of column k of P^m (ignoring leading zeros) equals: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} L(q^j*x) where L(x) satisfies: x/(1-x) = Sum_{n>=1} Product_{j=0..n-1} L(q^j*x)/(j+1) and L(x) equals the g.f. of column 0 of the matrix log of P (A111839).

Let q=8; the g.f. of column k of matrix power P^m is:

1 + (m*q^k)*L(x) + (m*q^k)^2/2!*L(x)*L(q*x) +

(m*q^k)^3/3!*L(x)*L(q*x)*L(q^2*x) +

(m*q^k)^4/4!*L(x)*L(q*x)*L(q^2*x)*L(q^3*x) + ...

where L(x) satisfies:

x/(1-x) = L(x) + L(x)*L(q*x)/2! + L(x)*L(q*x)*L(q^2*x)/3! + ...

and L(x) = x - 6/2!*x^2 + 142/3!*x^3 + 31800/4!*x^4 +... (A111839).

Thus the g.f. of column 0 of matrix power P^m is:

1 + m*L(x) + m^2/2!*L(x)*L(8*x) + m^3/3!*L(x)*L(8*x)*L(8^2*x) + m^4/4!*L(x)*L(8*x)*L(8^2*x)*L(8^3*x) + ...

Triangle P begins:

1;

1,1;

1,8,1;

1,232,64,1;

1,36968,16192,512,1;

1,35593832,21928768,1047040,4096,1;

1,219379963496,178379459392,11424946688,67096576,32768,1; ...

where P^8 shifts columns left and up one place:

1;

8,1;

232,64,1;

36968,16192,512,1; ...

(PARI) {P(n, k, q=8)=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|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(A[n+1, k+1]))}

Cf. A111836 (column 1), A111837 (row sums), A111838 (matrix log); triangles: A110503 (q=-1), A078121 (q=2), A078122 (q=3), A078536 (q=4), A111820 (q=5), A111825 (q=6), A111830 (q=7).

Sequence in context: A022171 A015121 A156766 this_sequence A010154 A109011 A019763

Adjacent sequences: A111832 A111833 A111834 this_sequence A111836 A111837 A111838

nonn,tabl

Gottfried Helms (helms(AT)uni-kassel.de) and Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2005