0,4

Column 0 equals A003319 (indecomposable permutations). Amazingly, column 1 (A104981) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A104986.

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 17 2009: (Start)

Square array A156628 has columns found in this triangle T:

Column 0 of A156628 = column 0 of T = A003319;

Column 1 of A156628 = column 1 of T = A104981;

Column 2 of A156628 = column 2 of T = A003319 shifted;

Column 3 of A156628 = column 1 of T^2 (A104988);

Column 5 of A156628 = column 2 of T^2 (A104988). (End)

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

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

SHIFT_LEFT(column 0 of T^1) = 1*(column 2 of T);

SHIFT_LEFT(column 0 of T^2) = 2*(column 3 of T);

where SHIFT_LEFT of column sequence shifts 1 place left.

Triangle T begins:

1;

1,1;

3,2,1;

13,7,3,1;

71,33,13,4,1;

461,191,71,21,5,1;

3447,1297,461,133,31,6,1;

29093,10063,3447,977,225,43,7,1;

273343,87669,29093,8135,1859,353,57,8,1;

2829325,847015,273343,75609,17185,3251,523,73,9,1; ...

Matrix inverse T^-1 is A104984 which begins:

1;

-1,1;

-1,-2,1;

-3,-1,-3,1;

-13,-3,-1,-4,1;

-71,-13,-3,-1,-5,1;

-461,-71,-13,-3,-1,-6,1; ...

Matrix T also satisfies:

[I + SHIFT_LEFT(T)] = [I - SHIFT_DOWN(T)]^-1, which starts:

1;

1,1;

2,1,1;

7,3,1,1;

33,13,4,1,1;

191,71,21,5,1,1; ...

where SHIFT_DOWN(T) shifts columns of T down 1 row,

and SHIFT_LEFT(T) shifts rows of T left 1 column,

with both operations leaving zeros in the diagonal.

(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, 0)*T(n, j+k+1)))))} (PARI) {T(n, k)=if(n<k|k<0, 0, (matrix(n+1, n+1, m, j, if(m==j, 1, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^-1)[n+1, k+1])}

Cf. A003319 (column 0), A104981 (column 1), A104983 (row sums), A104984 (matrix inverse), A104988 (matrix square), A104990 (matrix cube), A104986 (matrix log).

Cf. A156628. [From Paul D. Hanna (pauldhanna(AT)juno.com), Feb 17 2009]

Sequence in context: A048647 A059438 A156628 this_sequence A134090 A132845 A129652

Adjacent sequences: A104977 A104978 A104979 this_sequence A104981 A104982 A104983

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 10 2005