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%I #19 May 20 2021 04:44:55
%S 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 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 2,3,4,5,6,14,64,171,512,2047,1,2,3,4,5,6,8,36,136,341,1024,4095
%N 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 Seiichi Manyama, <a href="/A307078/b307078.txt">Antidiagonals n = 0..139, flattened</a>
%F A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).
%F A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 3, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 7, 4, 3, 3, 3, 3, 3, 3, 3, ...
%e 15, 8, 5, 4, 4, 4, 4, 4, 4, ...
%e 31, 16, 10, 6, 5, 5, 5, 5, 5, ...
%e 63, 32, 21, 12, 7, 6, 6, 6, 6, ...
%e 127, 64, 43, 28, 14, 8, 7, 7, 7, ...
%e 255, 128, 86, 64, 36, 16, 9, 8, 8, ...
%e 511, 256, 171, 136, 93, 45, 18, 10, 9, ...
%t 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 Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.
%Y Cf. A306846, A306915, A307079.
%K nonn,tabl
%O 0,3
%A _Seiichi Manyama_, Mar 22 2019
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