0,4

Column 0 equals A111529 (related to log of factorial series). Column 2 (A111538) equals SHIFT_LEFT(column 0 of LOG(T)), where the matrix logarithm, LOG(T), equals the integer matrix A111541.

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.

SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);

SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);

SHIFT_LEFT(column 0 of LOG(T)) = column 2 of T;

SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);

SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1,1;

4,2,1;

22,8,3,1;

148,44,14,4,1;

1156,296,84,22,5,1;

10192,2312,600,148,32,6,1;

99688,20384,4908,1156,242,44,7,1;

1069168,199376,44952,10192,2084,372,58,8,1; ...

After initial term, column 1 is twice column 0.

Matrix inverse T^-1 = A111540 starts:

1;

-1,1;

-2,-2,1;

-8,-2,-3,1;

-44,-8,-2,-4,1;

-296,-44,-8,-2,-5,1;

-2312,-296,-44,-8,-2,-6,1; ...

where columns are all equal after initial terms;

compare columns of T^-1 to column 1 of T.

Matrix logarithm LOG(T) = A111541 is:

0;

1,0;

3,2,0;

14,5,3,0;

84,22,8,4,0;

600,128,36,12,5,0;

4908,896,212,58,17,6,0; ...

compare column 0 of LOG(T) to column 2 of T.

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k, 1, if(n==k+1, n, k*T(n, k+1)+sum(j=0, n-k-1, T(j+1, 1)*T(n, j+k+1)))))}

Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.

Sequence in context: A049429 A109244 A039948 this_sequence A111559 A105623 A138271

Adjacent sequences: A111533 A111534 A111535 this_sequence A111537 A111538 A111539

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 06 2005