A104988 Matrix square of triangle A104980.

1, 2, 1, 8, 4, 1, 42, 20, 6, 1, 266, 120, 38, 8, 1, 1954, 836, 270, 62, 10, 1, 16270, 6616, 2150, 516, 92, 12, 1, 151218, 58576, 19030, 4688, 882, 128, 14, 1, 1551334, 573672, 185674, 46516, 9050, 1392, 170, 16, 1, 17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 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) = 2*A104980(n+3, 3) = 2*A104982(n) for n>=0.

Example

Triangle begins:
         1;
         2,       1;
         8,       4,       1;
        42,      20,       6,      1;
       266,     120,      38,      8,     1;
      1954,     836,     270,     62,    10,     1;
     16270,    6616,    2150,    516,    92,    12,    1;
    151218,   58576,   19030,   4688,   882,   128,   14,   1;
   1551334,  573672,  185674,  46516,  9050,  1392,  170,  16,  1;
  17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 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, 2][[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))))^-2)[n+1, k+1])

Crossrefs

Cf. A104980, A104982 (column 0), A104989 (row sums).

Sequence in context: A110446 A109979 A110171 * A343296 A136225 A341724

Adjacent sequences: A104985 A104986 A104987 * A104989 A104990 A104991

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 10 2005

Status

approved