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A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

%I A307078 #19 May 20 2021 04:44:55
%S A307078 1,1,3,1,2,7,1,2,4,15,1,2,3,8,31,1,2,3,5,16,63,1,2,3,4,10,32,127,1,2,
%T A307078 3,4,6,21,64,255,1,2,3,4,5,12,43,128,511,1,2,3,4,5,7,28,86,256,1023,1,
%U A307078 2,3,4,5,6,14,64,171,512,2047,1,2,3,4,5,6,8,36,136,341,1024,4095
%N A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).
%H A307078 Seiichi Manyama, <a href="/A307078/b307078.txt">Antidiagonals n = 0..139, flattened</a>
%F A307078 A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).
%F A307078 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).
%e A307078 Square array begins:
%e A307078      1,   1,   1,   1,  1,  1,  1,  1, 1, ...
%e A307078      3,   2,   2,   2,  2,  2,  2,  2, 2, ...
%e A307078      7,   4,   3,   3,  3,  3,  3,  3, 3, ...
%e A307078     15,   8,   5,   4,  4,  4,  4,  4, 4, ...
%e A307078     31,  16,  10,   6,  5,  5,  5,  5, 5, ...
%e A307078     63,  32,  21,  12,  7,  6,  6,  6, 6, ...
%e A307078    127,  64,  43,  28, 14,  8,  7,  7, 7, ...
%e A307078    255, 128,  86,  64, 36, 16,  9,  8, 8, ...
%e A307078    511, 256, 171, 136, 93, 45, 18, 10, 9, ...
%t A307078 T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)
%Y A307078 Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.
%Y A307078 Cf. A306846, A306915, A307079.
%K A307078 nonn,tabl
%O A307078 0,3
%A A307078 _Seiichi Manyama_, Mar 22 2019