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:
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 10 2005
STATUS
approved