A395078 a(n) = Sum_{j=0..n} binomial(n, j) * j^(n + j) * (1 - j)^(n - j).

1, 1, 16, 633, 48424, 6094625, 1143319656, 299199471697, 104166748496560, 46559005832738529, 25986265122668944240, 17713822771819911006281, 14481759761730572172382104, 13986094497715126301126461057, 15753377133338198042362649125528, 20468370147831373470211360930072065

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..200

Formula

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

Maple

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

Mathematica

A395078[n_] := If[n <= 1, 1, Sum[Binomial[n, j]*j^(n+j)*(1-j)^(n-j), {j, 2, n}]];
Array[A395078, 20, 0] (* Paolo Xausa, Jun 05 2026 *)

Prog

(Python) # Using the recurrence of Mikhail Kurkov in A394825.
def aList(lim: int) -> list[int]:
    result = [0] * (lim + 1)
    result[0] = 1
    row = [1] * (lim + 1)
    for k in range(1, lim + 1): row[k] += k * row[k - 1]
    for i in range(1, lim + 1):
        old = row[0]
        row[0] = 0
        for k in range(1, lim + 1):
            row[k], old = k * (row[k - 1] + row[k] - old), row[k]
        result[i] = row[i]
    return result
print(aList(15))

Crossrefs

Main diagonal of A394825.

Cf. A226775.

Sequence in context: A393877 A091496 A139166 * A128048 A197670 A162546

Adjacent sequences: A395075 A395076 A395077 * A395079 A395080 A395081

Keyword

nonn

Author

Peter Luschny, Apr 16 2026

Status

approved