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%I #30 Jan 09 2024 12:33:42
%S 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,
%T 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,
%U 71,47,1,0,0,1,3,7,15,31,57,99,131,76,1,0
%N 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).
%e n\k[0][1][2][3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
%e [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e [1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e [2] 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322 [A000032]
%e [3] 0, 1, 3, 7, 11, 21, 39, 71, 131, 241, 443, 815, 1499 [A001644]
%e [4] 0, 1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631 [A073817]
%e [5] 0, 1, 3, 7, 15, 31, 57, 113, 223, 439, 863, 1695, 3333 [A074048]
%e [6] 0, 1, 3, 7, 15, 31, 63, 120, 239, 475, 943, 1871, 3711 [A074584]
%e [7] 0, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903 [A104621]
%e [8] 0, 1, 3, 7, 15, 31, 63, 127, 255, 502, 1003, 2003, 3999 [A105754]
%e [.] . . . . . . . . . . . . .
%e oo] 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095 [A000225]
%e '
%e As a triangular array, starts:
%e 0,
%e 0, 0,
%e 0, 1, 0,
%e 0, 1, 1, 0,
%e 0, 1, 3, 1, 0,
%e 0, 1, 3, 4, 1, 0,
%e 0, 1, 3, 7, 7, 1, 0,
%e 0, 1, 3, 7, 11, 11, 1, 0,
%e 0, 1, 3, 7, 15, 21, 18, 1, 0,
%e 0, 1, 3, 7, 15, 26, 39, 29, 1, 0,
%p A := proc(n, k) f := -log((1+add((-1)^(j+1)*x^j, j=1..n))^(-1));
%p (-1)^(k+1)*k*coeff(series(f,x,k+2),x,k) end:
%p seq(print(seq(A(n,k), k=0..12)), n=0..8);
%t 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}]];
%t Table[A[n-k, k], {n, 0, 12}, {k, 0, n}] (* _Jean-François Alcover_, Jun 28 2019, from Maple *)
%Y Cf. A247506, A000225, A000032, A001644, A073817, A074048, A074584, A104621, A105754.
%Y Cf. A125127.
%K tabl,nonn
%O 0,13
%A _Peter Luschny_, Nov 02 2014
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