A104990 Matrix cube of triangle A104980.

1, 3, 1, 15, 6, 1, 93, 39, 9, 1, 675, 285, 75, 12, 1, 5577, 2331, 657, 123, 15, 1, 51555, 21153, 6207, 1269, 183, 18, 1, 526809, 211227, 63549, 13743, 2181, 255, 21, 1, 5895819, 2304321, 704319, 158325, 26739, 3453, 339, 24, 1, 71733585, 27291843, 8424813, 1947711, 343641, 47355, 5145, 435, 27, 1

Offset: 0

Author

Paul D. Hanna, Apr 10 2005

Keywords

• nonn
• tabl

Comments

Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.

Examples

			Triangle begins:
         1;
         3,        1;
        15,        6,       1;
        93,       39,       9,       1;
       675,      285,      75,      12,      1;
      5577,     2331,     657,     123,     15,     1;
     51555,    21153,    6207,    1269,    183,    18,    1;
    526809,   211227,   63549,   13743,   2181,   255,   21,   1;
   5895819,  2304321,  704319,  158325,  26739,  3453,  339,  24,  1;
  71733585, 27291843, 8424813, 1947711, 343641, 47355, 5145, 435, 27, 1;
		

Links

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

Crossrefs

Cf. A104980, A104982 (column 0), A104991 (row sums).

Programs

Mathematica

t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*t[n, k+1] + Sum[t[j, 0]*t[n, j+k+1], {j, 0, n-k-1}]]]]; (* t = A104980 *)
M:= With[{q=20}, Table[If[j>i, 0, t[i, j]], {i,0,q}, {j,0,q}]];
Table[MatrixPower[M, 3][[n+1, k+1]], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2021 *)

PARI

T(n,k)=if(n
				

Formula

T(n+1, 0) = 3*A104980(n+4, 4) for n>=0.