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.

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, 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, 129, 731, 1268, 1923, 1281, 376, 55, 6

Offset

0,6

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Example

Square array begins:
   1,  1,   1,    1,     1,      1,       1, ...
   1,  1,   1,    1,     1,      1,       1, ...
   2,  2,   2,    2,     2,      2,       2, ...
   2,  3,   5,    9,    17,     33,      65, ...
   3,  5,  11,   29,    83,    245,     731, ...
   3,  8,  26,   92,   338,   1268,    4826, ...
   4, 13,  63,  343,  1923,  10903,   62283, ...
   4, 21, 153, 1281, 11553, 108801, 1050753, ...

Mathematica

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 *)

Crossrefs

Columns 0-5 give A004526(n+2), A000045(n+1), A051286, A181545, A181546, A181547.

Rows 0-5 give A000012, A000012, A007395, A000051, A168607, A074506.

Main diagonal gives A323769.

Cf. A011973,

Sequence in context: A110537 A144434 A322057 * A357824 A159936 A305749

Adjacent sequences: A323764 A323765 A323766 * A323768 A323769 A323770

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 27 2019

Status

approved