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A247505
Generalized Lucas numbers: square array A(n,k) read by antidiagonals, A(n,k)=(-1)^(k+1)*k*[x^k](-log((1+sum_{j=1..n}(-1)^(j+1)*x^j)^(-1))), (n>=0, k>=0).
3
0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 3, 4, 1, 0, 0, 1, 3, 7, 7, 1, 0, 0, 1, 3, 7, 11, 11, 1, 0, 0, 1, 3, 7, 15, 21, 18, 1, 0, 0, 1, 3, 7, 15, 26, 39, 29, 1, 0, 0, 1, 3, 7, 15, 31, 51, 71, 47, 1, 0, 0, 1, 3, 7, 15, 31, 57, 99, 131, 76, 1, 0
OFFSET
0,13
EXAMPLE
n\k[0][1][2][3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
[1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
[2] 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322 [A000032]
[3] 0, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499 [A001644]
[4] 0, 1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631 [A073817]
[5] 0, 1, 3, 7, 15, 31, 57, 113, 223, 439, 863, 1695, 3333 [A074048]
[6] 0, 1, 3, 7, 15, 31, 63, 120, 239, 475, 943, 1871, 3711 [A074584]
[7] 0, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903 [A104621]
[8] 0, 1, 3, 7, 15, 31, 63, 127, 255, 502, 1003, 2003, 3999 [A105754]
[.] . . . . . . . . . . . . .
oo] 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095 [A000225]
'
As a triangular array, starts:
0,
0, 0,
0, 1, 0,
0, 1, 1, 0,
0, 1, 3, 1, 0,
0, 1, 3, 4, 1, 0,
0, 1, 3, 7, 7, 1, 0,
0, 1, 3, 7, 11, 11, 1, 0,
0, 1, 3, 7, 15, 21, 18, 1, 0,
0, 1, 3, 7, 15, 26, 39, 29, 1, 0,
MAPLE
A := proc(n, k) f := -log((1+add((-1)^(j+1)*x^j, j=1..n))^(-1));
(-1)^(k+1)*k*coeff(series(f, x, k+2), x, k) end:
seq(print(seq(A(n, k), k=0..12)), n=0..8);
MATHEMATICA
A[n_, k_] := Module[{f, x}, f = -Log[(1+Sum[(-1)^(j+1) x^j, {j, 1, n}] )^(-1)]; (-1)^(k+1) k SeriesCoefficient[f, {x, 0, k}]];
Table[A[n-k, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019, from Maple *)
KEYWORD
tabl,nonn
AUTHOR
Peter Luschny, Nov 02 2014
STATUS
approved