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A104986
Matrix logarithm of triangle A104980.
5
0, 1, 0, 2, 2, 0, 7, 4, 3, 0, 33, 14, 7, 4, 0, 191, 66, 27, 11, 5, 0, 1297, 382, 137, 48, 16, 6, 0, 10063, 2594, 843, 270, 79, 22, 7, 0, 87669, 20126, 6041, 1820, 495, 122, 29, 8, 0, 847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0, 8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0
OFFSET
0,4
COMMENTS
Column 0 equals column 1 of triangular matrix A104980, which satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0. Column 1 equals twice column 0.
FORMULA
T(n, 0) = A104981(n), T(n+1, 1) = 2*T(n, 0) for n>=0.
EXAMPLE
Triangle begins:
0;
1, 0;
2, 2, 0;
7, 4, 3, 0;
33, 14, 7, 4, 0;
191, 66, 27, 11, 5, 0;
1297, 382, 137, 48, 16, 6, 0;
10063, 2594, 843, 270, 79, 22, 7, 0;
87669, 20126, 6041, 1820, 495, 122, 29, 8, 0;
847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0;
8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0; ...
MATHEMATICA
nmax = 10;
M = Table[If[n == k, 0, If[n == k+1, -n+1, -Coefficient[(1-1/Sum[i! x^i, {i, 0, n}])/x + O[x]^n, x, n-k-1]]], {n, 1, nmax+1}, {k, 1, nmax+1}];
T[n_, k_] /; 0 <= k <= n := Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p, 1, n+1}]; T[_, _] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, sum(p=1, n+1, (-1)^p*(matrix(n+1, n+1, m, j, if(m==j, 0, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^p)[n+1, k+1]/p))
CROSSREFS
Cf. A104980, A104981 (column 0), A104987 (row sums).
Sequence in context: A323675 A243492 A086118 * A060007 A021457 A305605
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 10 2005
STATUS
approved