A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1.

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

Offset

0,4

Comments

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.

From Paul D. Hanna, 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)

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

Formula

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.

Example

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.

Mathematica

T[n_, k_]:= T[n, k]= If[n<k || k<0, 0, If[n==k, 1, If[n==k+1, n, k T[n, k+1] + Sum[T[j, 0] T[n, j+k+1], {j, 0, n-k-1}]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)

Prog

(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)))))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(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])}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(Sage)
@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==n-1): return n
    else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 07 2021

Crossrefs

Cf. A003319 (column 0), A104981 (column 1), A104983 (row sums), A104984 (matrix inverse), A104988 (matrix square), A104990 (matrix cube), A104986 (matrix log), A156628.

Sequence in context: A180190 A059438 A156628 * A316566 A134090 A132845

Adjacent sequences: A104977 A104978 A104979 * A104981 A104982 A104983

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 10 2005

Status

approved