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A306713 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+1)).

%I A306713 #22 Jun 21 2021 03:01:13
%S A306713 1,1,1,1,0,2,1,0,1,3,1,0,0,1,5,1,0,0,1,1,8,1,0,0,0,1,2,13,1,0,0,0,1,0,
%T A306713 2,21,1,0,0,0,0,1,1,3,34,1,0,0,0,0,1,0,2,4,55,1,0,0,0,0,0,1,0,1,5,89,
%U A306713 1,0,0,0,0,0,1,0,1,1,7,144,1,0,0,0,0,0,0,1,0,2,3,9,233
%N A306713 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+1)).
%C A306713 A(n,k) is the number of compositions of n into parts k and k+1.
%H A306713 Seiichi Manyama, <a href="/A306713/b306713.txt">Antidiagonals n = 0..139, flattened</a>
%F A306713 A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).
%e A306713 Square array begins:
%e A306713     1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A306713     1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A306713     2, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e A306713     3, 1, 1, 0, 0, 0, 0, 0, 0, ...
%e A306713     5, 1, 1, 1, 0, 0, 0, 0, 0, ...
%e A306713     8, 2, 0, 1, 1, 0, 0, 0, 0, ...
%e A306713    13, 2, 1, 0, 1, 1, 0, 0, 0, ...
%e A306713    21, 3, 2, 0, 0, 1, 1, 0, 0, ...
%e A306713    34, 4, 1, 1, 0, 0, 1, 1, 0, ...
%e A306713    55, 5, 1, 2, 0, 0, 0, 1, 1, ...
%t A306713 T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 21 2021 *)
%Y A306713 Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.
%Y A306713 Cf. A306646, A306680.
%K A306713 nonn,tabl
%O A306713 0,6
%A A306713 _Seiichi Manyama_, Mar 05 2019