A323769 -id:A323769 - cp's OEIS frontend

A323768 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^k.

1, 1, 2, 3, 5, 14, 43, 171, 1234, 9075, 94295, 1685324, 28688843, 804627839, 34189166176, 1379425012899, 106952499421507, 10394354507270548, 1052079100669253203, 221582922117645427461, 48152920476428200426258, 13152336142340905111739041

Offset: 0

Vaclav Kotesovec, Jan 27 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..123

Cf. A323769, A295612.

Table[Sum[Binomial[n-k, k]^k, {k, 0, n/2}], {n, 0, 25}]

{a(n) = sum(k=0, n\2, binomial(n-k, k)^k)} \\ Seiichi Manyama, Jan 27 2019

Limit_{n->infinity} a(n)^(1/n^2) = ((1-r)/r)^(r^2/(4*r-1)) = 1.17123387669321050316385592324128471190583619526359450226558587879190245..., where r = A323773 = 0.3663201503052830964087236563781171194011826607210994595... is the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

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.

Original entry on oeis.org

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

Seiichi Manyama, Jan 27 2019

			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, ...

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

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,

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 *)

cp's OEIS Frontend

A323768 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^k.

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