A395078 - OEIS

%I #14 Jun 05 2026 12:50:04

%S 1,1,16,633,48424,6094625,1143319656,299199471697,104166748496560,

%T 46559005832738529,25986265122668944240,17713822771819911006281,

%U 14481759761730572172382104,13986094497715126301126461057,15753377133338198042362649125528,20468370147831373470211360930072065

%N a(n) = Sum_{j=0..n} binomial(n, j) * j^(n + j) * (1 - j)^(n - j).

%H Paolo Xausa, <a href="/A395078/b395078.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ 2^(2*n - 1/2) * n^(2*n) / (sqrt(1-w) * exp(2*n-1) * w^(n - 1/2) * (2-w)^n), where w = -LambertW(-2*exp(-2)) = -A226775. - _Vaclav Kotesovec_, Jun 05 2026

%p a := n -> local j; add(binomial(n, j) * j^(n + j) * (1 - j)^(n - j), j = 0..n): seq(a(n), n = 0..16);

%t A395078[n_] := If[n <= 1, 1, Sum[Binomial[n, j]*j^(n+j)*(1-j)^(n-j), {j, 2, n}]];

%t Array[A395078, 20, 0] (* _Paolo Xausa_, Jun 05 2026 *)

%o (Python) # Using the recurrence of _Mikhail Kurkov_ in A394825.

%o def aList(lim: int) -> list[int]:

%o     result = [0] * (lim + 1)

%o     result[0] = 1

%o     row = [1] * (lim + 1)

%o     for k in range(1, lim + 1): row[k] += k * row[k - 1]

%o     for i in range(1, lim + 1):

%o         old = row[0]

%o         row[0] = 0

%o         for k in range(1, lim + 1):

%o             row[k], old = k * (row[k - 1] + row[k] - old), row[k]

%o         result[i] = row[i]

%o     return result

%o print(aList(15))

%Y Main diagonal of A394825.

%Y Cf. A226775.

%K nonn

%O 0,3

%A _Peter Luschny_, Apr 16 2026