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A307393 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 A307393 #25 May 20 2021 04:44:33
%S A307393 1,1,5,1,4,16,1,4,11,42,1,4,10,26,99,1,4,10,21,57,219,1,4,10,20,42,
%T A307393 120,466,1,4,10,20,36,84,247,968,1,4,10,20,35,64,169,502,1981,1,4,10,
%U A307393 20,35,57,120,340,1013,4017,1,4,10,20,35,56,93,240,682,2036,8100
%N A307393 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 A307393 Seiichi Manyama, <a href="/A307393/b307393.txt">Antidiagonals n = 0..139, flattened</a>
%F A307393 A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3).
%F A307393 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).
%e A307393 Square array begins:
%e A307393      1,   1,   1,   1,   1,   1,   1,   1, ...
%e A307393      5,   4,   4,   4,   4,   4,   4,   4, ...
%e A307393     16,  11,  10,  10,  10,  10,  10,  10, ...
%e A307393     42,  26,  21,  20,  20,  20,  20,  20, ...
%e A307393     99,  57,  42,  36,  35,  35,  35,  35, ...
%e A307393    219, 120,  84,  64,  57,  56,  56,  56, ...
%e A307393    466, 247, 169, 120,  93,  85,  84,  84, ...
%e A307393    968, 502, 340, 240, 165, 130, 121, 120, ...
%t A307393 T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y A307393 Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3).
%Y A307393 Cf. A306915, A306846, A307078, A307394.
%K A307393 nonn,tabl
%O A307393 0,3
%A A307393 _Seiichi Manyama_, Apr 07 2019