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A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

%I A307079 #25 May 20 2021 04:44:47
%S A307079 1,1,1,1,2,1,1,2,2,1,1,2,3,0,1,1,2,3,3,-4,1,1,2,3,4,0,-8,1,1,2,3,4,4,
%T A307079 -9,-8,1,1,2,3,4,5,0,-27,0,1,1,2,3,4,5,5,-14,-54,16,1,1,2,3,4,5,6,0,
%U A307079 -48,-81,32,1,1,2,3,4,5,6,6,-20,-116,-81,32,1
%N A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).
%H A307079 Seiichi Manyama, <a href="/A307079/b307079.txt">Antidiagonals n = 0..139, flattened</a>
%F A307079 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+1,k*j+1).
%F A307079 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,k*j) * binomial(n-i,k*j).
%e A307079 Square array begins:
%e A307079    1,  1,   1,    1,   1,   1, 1, 1, 1, ...
%e A307079    1,  2,   2,    2,   2,   2, 2, 2, 2, ...
%e A307079    1,  2,   3,    3,   3,   3, 3, 3, 3, ...
%e A307079    1,  0,   3,    4,   4,   4, 4, 4, 4, ...
%e A307079    1, -4,   0,    4,   5,   5, 5, 5, 5, ...
%e A307079    1, -8,  -9,    0,   5,   6, 6, 6, 6, ...
%e A307079    1, -8, -27,  -14,   0,   6, 7, 7, 7, ...
%e A307079    1,  0, -54,  -48, -20,   0, 7, 8, 8, ...
%e A307079    1, 16, -81, -116, -75, -27, 0, 8, 9, ...
%t A307079 T[n_, k_] := Sum[(-1)^j * Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y A307079 Columns 1-6 give A000012, A099087, A057682(n+1), A099587(n+1), A289321(n+1), A307089.
%Y A307079 Cf. A306914, A307039, A307078.
%K A307079 sign,tabl,look
%O A307079 0,5
%A A307079 _Seiichi Manyama_, Mar 22 2019