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Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).
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%I #9 Mar 27 2018 16:24:58

%S 1,1,0,1,1,0,1,1,2,0,1,1,4,6,0,1,1,4,18,24,0,1,1,4,24,120,120,0,1,1,4,

%T 24,168,960,720,0,1,1,4,24,192,1560,9360,5040,0,1,1,4,24,192,1800,

%U 17280,105840,40320,0,1,1,4,24,192,1920,20880,221760,1370880,362880,0

%N Square array read by antidiagonals, A(n,k) = k!*[x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0,k>=0).

%e n\k[0][1][2] [3] [4] [5] [6] [7] [8] [9]

%e [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

%e [1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880 [A000142]

%e [2] 1, 1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400 [A005442]

%e [3] 1, 1, 4, 24, 168, 1560, 17280, 221760, 3265920, 54069120

%e [.] . . . . . . . . . . . . .

%e oo] 1, 1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280 [A002866]

%e '

%e As a triangular array, starts:

%e 1,

%e 1, 0,

%e 1, 1, 0,

%e 1, 1, 2, 0,

%e 1, 1, 4, 6, 0,

%e 1, 1, 4, 18, 24, 0,

%e 1, 1, 4, 24, 120, 120, 0,

%e 1, 1, 4, 24, 168, 960, 720, 0.

%p A := (n,k) -> k!*coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k): seq(print(seq(A(n,k), k=0..9)), n=0..7);

%Y Cf. A247505, A247506, A002866, A000142, A005442.

%K nonn,tabl

%O 0,9

%A _Peter Luschny_, Nov 03 2014