login
A104988
Matrix square of triangle A104980.
5
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 10 2005
STATUS
approved