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A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

%I A306680 #34 Jun 21 2021 03:01:09
%S A306680 1,1,2,1,1,3,1,1,2,4,1,1,1,3,5,1,1,1,2,5,6,1,1,1,1,4,8,7,1,1,1,1,2,7,
%T A306680 13,8,1,1,1,1,1,5,12,21,9,1,1,1,1,1,2,11,21,34,10,1,1,1,1,1,1,6,21,37,
%U A306680 55,11,1,1,1,1,1,1,2,16,37,65,89,12
%N A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).
%H A306680 Seiichi Manyama, <a href="/A306680/b306680.txt">Antidiagonals n = 0..139, flattened</a>
%F A306680 A(n,k) = Sum_{j=0..n} binomial(n-j,k*j).
%F A306680 A(n,k) = A306713(k*n,k) for k > 0.
%e A306680 A(4,1) = A306713(4,1) = 5, A(4,2) = A306713(8,2) = 4.
%e A306680 Square array begins:
%e A306680    1,  1,  1,  1,  1,  1, 1, 1, 1, ...
%e A306680    2,  1,  1,  1,  1,  1, 1, 1, 1, ...
%e A306680    3,  2,  1,  1,  1,  1, 1, 1, 1, ...
%e A306680    4,  3,  2,  1,  1,  1, 1, 1, 1, ...
%e A306680    5,  5,  4,  2,  1,  1, 1, 1, 1, ...
%e A306680    6,  8,  7,  5,  2,  1, 1, 1, 1, ...
%e A306680    7, 13, 12, 11,  6,  2, 1, 1, 1, ...
%e A306680    8, 21, 21, 21, 16,  7, 2, 1, 1, ...
%e A306680    9, 34, 37, 37, 36, 22, 8, 2, 1, ...
%t A306680 T[n_, k_] := Sum[Binomial[n - j, k*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 21 2021 *)
%Y A306680 Columns 0-9 give A000027(n+1), A000045(n+1), A005251(n+1), A003522, A005676, A099132, A293169, A306721, A306752, A306753.
%Y A306680 Cf. A306646, A306713, A306735.
%K A306680 nonn,tabl,look
%O A306680 0,3
%A A306680 _Seiichi Manyama_, Mar 05 2019