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A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).

%I A306646 #31 Jun 21 2021 03:01:01
%S A306646 2,3,1,4,0,3,5,0,2,4,6,0,0,3,7,7,0,0,3,2,11,8,0,0,0,4,5,18,9,0,0,0,4,
%T A306646 0,5,29,10,0,0,0,0,5,3,7,47,11,0,0,0,0,5,0,7,10,76,12,0,0,0,0,0,6,0,4,
%U A306646 12,123,13,0,0,0,0,0,6,0,4,3,17,199
%N A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).
%H A306646 Seiichi Manyama, <a href="/A306646/b306646.txt">Antidiagonals n = 0..139, flattened</a>
%F A306646 A(0,k) = k+1 and A(n,k) = n*Sum_{j=1..floor(n/k)} binomial(j,n-k*j)/j for n > 0.
%F A306646 A(n,k) = (k+1)*A306713(n,k) - A306713(n-k,k) for n >= k.
%e A306646 A(6,1) = 6*Sum_{j=1..6} binomial(j,6-j)/j = 6*(1/3+3/2+1+1/6) = 18.
%e A306646 A(6,2) = 6*Sum_{j=1..3} binomial(j,6-2*j)/j = 6*(1/2+1/3) = 5.
%e A306646 Square array begins:
%e A306646     2,  3, 4, 5, 6, 7, 8, 9, 10, 11, ...
%e A306646     1,  0, 0, 0, 0, 0, 0, 0,  0,  0, ...
%e A306646     3,  2, 0, 0, 0, 0, 0, 0,  0,  0, ...
%e A306646     4,  3, 3, 0, 0, 0, 0, 0,  0,  0, ...
%e A306646     7,  2, 4, 4, 0, 0, 0, 0,  0,  0, ...
%e A306646    11,  5, 0, 5, 5, 0, 0, 0,  0,  0, ...
%e A306646    18,  5, 3, 0, 6, 6, 0, 0,  0,  0, ...
%e A306646    29,  7, 7, 0, 0, 7, 7, 0,  0,  0, ...
%e A306646    47, 10, 4, 4, 0, 0, 8, 8,  0,  0, ...
%e A306646    76, 12, 3, 9, 0, 0, 0, 9,  9,  0, ...
%t A306646 T[0, k_] := k + 1; T[n_, k_] := n *Sum[Binomial[j, n - k*j]/j, {j, 1, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 21 2021 *)
%Y A306646 Columns 1-9 give A000032, A001608, A050443, A087935, A087936, A306755, A306756, A306757, A306758.
%Y A306646 Cf. A306713, A306735.
%K A306646 nonn,tabl
%O A306646 0,1
%A A306646 _Seiichi Manyama_, Mar 03 2019