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
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..123
Programs
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Mathematica
Table[Sum[Binomial[n-k, k]^k, {k, 0, n/2}], {n, 0, 25}] -
PARI
{a(n) = sum(k=0, n\2, binomial(n-k, k)^k)} \\ Seiichi Manyama, Jan 27 2019
Formula
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).