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A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).

%I A307394 #25 May 20 2021 04:44:39
%S A307394 1,1,3,1,4,6,1,4,9,10,1,4,10,14,15,1,4,10,19,15,21,1,4,10,20,28,8,28,
%T A307394 1,4,10,20,34,28,-7,36,1,4,10,20,35,48,1,-22,45,1,4,10,20,35,55,48,
%U A307394 -80,-21,55,1,4,10,20,35,56,75,0,-242,12,66,1,4,10,20,35,56,83,75,-164,-485,77,78
%N A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).
%H A307394 Seiichi Manyama, <a href="/A307394/b307394.txt">Antidiagonals n = 0..139, flattened</a>
%F A307394 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+3,k*j+3).
%F A307394 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).
%e A307394 Square array begins:
%e A307394     1,   1,    1,    1,  1,   1,   1,   1,   1, ...
%e A307394     3,   4,    4,    4,  4,   4,   4,   4,   4, ...
%e A307394     6,   9,   10,   10, 10,  10,  10,  10,  10, ...
%e A307394    10,  14,   19,   20, 20,  20,  20,  20,  20, ...
%e A307394    15,  15,   28,   34, 35,  35,  35,  35,  35, ...
%e A307394    21,   8,   28,   48, 55,  56,  56,  56,  56, ...
%e A307394    28,  -7,    1,   48, 75,  83,  84,  84,  84, ...
%e A307394    36, -22,  -80,    0, 75, 110, 119, 120, 120, ...
%e A307394    45, -21, -242, -164,  0, 110, 154, 164, 165, ...
%t A307394 T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y A307394 Columns 1-5 give A000217(n+1), A279230, A307395, A099589(n+3), A289388(n+3).
%Y A307394 Cf. A306914, A307039, A307079, A307393.
%K A307394 sign,tabl
%O A307394 0,3
%A A307394 _Seiichi Manyama_, Apr 07 2019