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A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

%I A306914 #36 Apr 08 2019 03:45:29
%S A306914 1,1,0,1,2,0,1,3,2,0,1,4,6,0,0,1,5,10,9,-4,0,1,6,15,20,9,-8,0,1,7,21,
%T A306914 35,34,0,-8,0,1,8,28,56,70,48,-27,0,0,1,9,36,84,126,125,48,-81,16,0,1,
%U A306914 10,45,120,210,252,200,0,-162,32,0
%N A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).
%H A306914 Seiichi Manyama, <a href="/A306914/b306914.txt">Antidiagonals n = 0..139, flattened</a>
%F A306914 A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+k-1,k*j+k-1).
%F A306914 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - _Seiichi Manyama_, Apr 07 2019
%e A306914 Square array begins:
%e A306914    1,  1,    1,    1,   1,    1,    1,    1, ...
%e A306914    0,  2,    3,    4,   5,    6,    7,    8, ...
%e A306914    0,  2,    6,   10,  15,   21,   28,   36, ...
%e A306914    0,  0,    9,   20,  35,   56,   84,  120, ...
%e A306914    0, -4,    9,   34,  70,  126,  210,  330, ...
%e A306914    0, -8,    0,   48, 125,  252,  462,  792, ...
%e A306914    0, -8,  -27,   48, 200,  461,  924, 1716, ...
%e A306914    0,  0,  -81,    0, 275,  780, 1715, 3432, ...
%e A306914    0, 16, -162, -164, 275, 1209, 2989, 6434, ...
%t A306914 A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
%t A306914 Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 20 2019 *)
%Y A306914 Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.
%Y A306914 Cf. A039912, A306913, A306915, A307039, A307079, A307394.
%K A306914 sign,tabl,look
%O A306914 0,5
%A A306914 _Seiichi Manyama_, Mar 16 2019