A307393 - OEIS

%I #25 May 20 2021 04:44:33

%S 1,1,5,1,4,16,1,4,11,42,1,4,10,26,99,1,4,10,21,57,219,1,4,10,20,42,

%T 120,466,1,4,10,20,36,84,247,968,1,4,10,20,35,64,169,502,1981,1,4,10,

%U 20,35,57,120,340,1013,4017,1,4,10,20,35,56,93,240,682,2036,8100

%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-4))/((1-x)^k-x^k).

%H Seiichi Manyama, <a href="/A307393/b307393.txt">Antidiagonals n = 0..139, flattened</a>

%F A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3).

%F A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).

%e Square array begins:

%e      1,   1,   1,   1,   1,   1,   1,   1, ...

%e      5,   4,   4,   4,   4,   4,   4,   4, ...

%e     16,  11,  10,  10,  10,  10,  10,  10, ...

%e     42,  26,  21,  20,  20,  20,  20,  20, ...

%e     99,  57,  42,  36,  35,  35,  35,  35, ...

%e    219, 120,  84,  64,  57,  56,  56,  56, ...

%e    466, 247, 169, 120,  93,  85,  84,  84, ...

%e    968, 502, 340, 240, 165, 130, 121, 120, ...

%t T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* _Amiram Eldar_, May 20 2021 *)

%Y Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3).

%Y Cf. A306915, A306846, A307078, A307394.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Apr 07 2019