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A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

%I A323767 #39 Nov 29 2020 02:11:56
%S A323767 1,1,1,1,1,2,1,1,2,2,1,1,2,3,3,1,1,2,5,5,3,1,1,2,9,11,8,4,1,1,2,17,29,
%T A323767 26,13,4,1,1,2,33,83,92,63,21,5,1,1,2,65,245,338,343,153,34,5,1,1,2,
%U A323767 129,731,1268,1923,1281,376,55,6
%N A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.
%H A323767 Seiichi Manyama, <a href="/A323767/b323767.txt">Antidiagonals n = 0..139, flattened</a>
%e A323767 Square array begins:
%e A323767    1,  1,   1,    1,     1,      1,       1, ...
%e A323767    1,  1,   1,    1,     1,      1,       1, ...
%e A323767    2,  2,   2,    2,     2,      2,       2, ...
%e A323767    2,  3,   5,    9,    17,     33,      65, ...
%e A323767    3,  5,  11,   29,    83,    245,     731, ...
%e A323767    3,  8,  26,   92,   338,   1268,    4826, ...
%e A323767    4, 13,  63,  343,  1923,  10903,   62283, ...
%e A323767    4, 21, 153, 1281, 11553, 108801, 1050753, ...
%t A323767 f := Sum[Power[Binomial[#1 - i, i], #2], {i, 0, #1/2}] &;a = Flatten[Reverse[DeleteCases[Table[Table[f[m - n, n], {n, 0, 20}], {m, 0, 20}], 0, Infinity], 2]] (* _Elijah Beregovsky_, Nov 24 2020 *)
%Y A323767 Columns 0-5 give A004526(n+2), A000045(n+1), A051286, A181545, A181546, A181547.
%Y A323767 Rows 0-5 give A000012, A000012, A007395, A000051, A168607, A074506.
%Y A323767 Main diagonal gives A323769.
%Y A323767 Cf. A011973,
%K A323767 nonn,tabl
%O A323767 0,6
%A A323767 _Seiichi Manyama_, Jan 27 2019