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A104990
Matrix cube of triangle A104980.
3
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 10 2005
STATUS
approved