0,4

Column 0 equals A111531 (related to log of factorial series). Column 4 (A111557) equals SHIFT_LEFT(column 0 of LOG(T)), where the matrix logarithm, LOG(T), equals the integer matrix A111560.

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+3, 3)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+4, 3) = 4*T(n+1, 0), T(n+5, 5) = T(n+1, 0), for n>=0.

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

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

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

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

SHIFT_LEFT(column 0 of LOG(T)) = column 4 of T;

SHIFT_LEFT(column 0 of T^1) = 1*(column 5 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1,1;

6,2,1;

46,10,3,1;

416,72,16,4,1;

4256,632,116,24,5,1;

48096,6352,1016,184,34,6,1;

591536,70912,10176,1664,282,46,7,1;

7840576,864192,113216,17024,2696,416,60,8,1; ...

After initial term, column 3 is 4 times column 0.

Matrix inverse T^-1 = A111559 starts:

1;

-1,1;

-4,-2,1;

-24,-4,-3,1;

-184,-24,-4,-4,1;

-1664,-184,-24,-4,-5,1;

-17024,-1664,-184,-24,-4,-6,1; ...

where columns are all equal after initial terms;

compare columns of T^-1 to column 3 of T.

Matrix logarithm LOG(T) = A111560 is:

0;

1,0;

5,2,0;

34,7,3,0;

282,44,10,4,0;

2696,354,60,14,5,0;

28792,3328,470,84,19,6,0; ...

compare column 0 of LOG(T) to column 4 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+3, 3)*T(n, j+k+1)))))}

Cf. A111531 (column 0), A111554 (column 1), A111555 (column 2), A111556 (column 3), A111557 (column 4), A111558 (row sums), A111559 (matrix inverse), A111560 (matrix log); related tables: A111528, A104980, A111536, A111544.

Adjacent sequences: A111550 A111551 A111552 this_sequence A111554 A111555 A111556

Sequence in context: A124913 A101818 A110321 this_sequence A068931 A061666 A136708

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 07 2005