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A306915 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).

%I A306915 #33 May 26 2021 00:55:19
%S A306915 1,1,2,1,2,4,1,3,4,8,1,4,6,8,16,1,5,10,11,16,32,1,6,15,20,21,32,64,1,
%T A306915 7,21,35,36,42,64,128,1,8,28,56,70,64,85,128,256,1,9,36,84,126,127,
%U A306915 120,171,256,512,1,10,45,120,210,252,220,240,342,512,1024
%N A306915 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).
%H A306915 Seiichi Manyama, <a href="/A306915/b306915.txt">Antidiagonals n = 0..139, flattened</a>
%F A306915 A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+k-1,k*j+k-1).
%F A306915 A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - _Seiichi Manyama_, Apr 07 2019
%e A306915 Square array begins:
%e A306915      1,   1,   1,   1,   1,    1,    1,    1, ...
%e A306915      2,   2,   3,   4,   5,    6,    7,    8, ...
%e A306915      4,   4,   6,  10,  15,   21,   28,   36, ...
%e A306915      8,   8,  11,  20,  35,   56,   84,  120, ...
%e A306915     16,  16,  21,  36,  70,  126,  210,  330, ...
%e A306915     32,  32,  42,  64, 127,  252,  462,  792, ...
%e A306915     64,  64,  85, 120, 220,  463,  924, 1716, ...
%e A306915    128, 128, 171, 240, 385,  804, 1717, 3432, ...
%e A306915    256, 256, 342, 496, 715, 1365, 3017, 6436, ...
%t A306915 A[n_, k_] := Sum[Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 25 2021 *)
%Y A306915 Columns (1+2),3-9 give A000079, A024495(n+2), A000749(n+3), A049016, A192080, A049017, A290995(n+7), A306939.
%Y A306915 Cf. A039912, A101508, A306846, A306913, A306914, A307047, A307078, A307393.
%K A306915 nonn,tabl
%O A306915 0,3
%A A306915 _Seiichi Manyama_, Mar 16 2019