A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.

1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 5, -11, 9, -4, 1, -17, 38, -33, 16, -5, 1, 71, -162, 145, -74, 25, -6, 1, -357, 824, -753, 396, -140, 36, -7, 1, 2101, -4892, 4535, -2434, 885, -237, 49, -8, 1, -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1, 108609, -255824, 241621, -133012, 50001, -13992, 3073, -548, 81

Offset: 0

Author

Paul D. Hanna, Dec 20 2004

Keywords

• sign
• tabl

Comments

Column 0 forms A101900. Absolute row sums form A101901.

Examples

			Rows begin:
        1;
       -1,     1;
        1,    -2,      1;
       -2,     4,     -3,     1;
        5,   -11,      9,    -4,     1;
      -17,    38,    -33,    16,    -5,    1;
       71,  -162,    145,   -74,    25,   -6,   1;
     -357,   824,   -753,   396,  -140,   36,  -7,   1,
     2101, -4892,   4535, -2434,   885, -237,  49,  -8,  1;
   -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1;
      ...
		

Links

• J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Crossrefs

Cf. A091351, A101900, A101901, A101898, A101899.

Programs

Mathematica

t[n_, k_] := t[n, k] = If[k>n || n<0 || k<0, 0, If[k==n, 1, If[k==0, -Sum[t[n, j], {j, 1, n}], Sum[t[n-k, j]*t[j+k-1, k-1], {j, 0, n-k}]]]]; Table[t[n ,k], {n,0,10}, {k, 0, n}] //Flatten (* Amiram Eldar, Nov 26 2018 *)

PARI

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

Formula

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