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

0,2

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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))))^-3)[n+1, k+1])

Crossrefs

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

Sequence in context: A092589 A048966 A297704 * A089463 A136231 A113389

Adjacent sequences: A104987 A104988 A104989 * A104991 A104992 A104993

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 10 2005

Status

approved