A362522 -id:A362522 - cp's OEIS frontend

A362523 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (k! * (n-3*k)!).

1, 1, 1, 7, 25, 61, 1201, 7771, 30577, 1058905, 9904321, 53722351, 2708688841, 33126146197, 228967340785, 15262865820931, 230517745701601, 1936173471789361, 161021598306402817, 2894434429492525015, 28614958982310290041

Offset: 0

Seiichi Manyama, Apr 23 2023

Seiichi Manyama, Table of n, a(n) for n = 0..427
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A362522.

Cf. A089464, A362348.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3))))

E.g.f.: exp(x - LambertW(-x^3)) = -LambertW(-x^3)/x^3 * exp(x).

a(n) ~ sqrt(3) * (exp(3*exp(-1/3)/2) + 2*cos(sqrt(3)*exp(-1/3)/2 - 2*Pi*n/3)) * n^(n-1) / exp(2*n/3 + exp(-1/3)/2 - 1). - Vaclav Kotesovec, Aug 05 2025

A361916 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (k+1)^(k-1) / (k! * (n-2*k)!).

Original entry on oeis.org

1, 1, -1, -5, 25, 161, -1409, -12221, 158705, 1733185, -30136769, -397326709, 8696945929, 134416055905, -3555479651905, -63044502191789, 1957884163020001, 39178556553643649, -1398250387206450305, -31169265056007817445, 1257498026543130033401

Offset: 0

Seiichi Manyama, Apr 24 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A105785, A361917, A362522.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2))))

E.g.f.: exp(x - LambertW(x^2)) = LambertW(x^2)/x^2 * exp(x).

A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 391, 2017, 20504, 139456, 1867681, 15751451, 262263442, 2638794094, 52589415971, 614628436801, 14274125637256, 190012483804952, 5041005195499849, 75288391385094811, 2246914521052963166, 37204717212894726706, 1233884675800841217847

Offset: 0

Seiichi Manyama, Apr 23 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A088957, A362525.

Cf. A362350, A362522.

Table[n!Sum[(k+1)^(k-1)/(2^k k!(n-2k)!),{k,0,Floor[n/2]}],{n,0,25}] (* Harvey P. Dale, Mar 30 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^2/2))))

E.g.f.: exp(x - LambertW(-x^2/2)) = -2 * LambertW(-x^2/2)/x^2 * exp(x).

a(n) ~ (exp(sqrt(2)*exp(-1/2) + 1) + (-1)^n*exp(1 - sqrt(2)*exp(-1/2))) * n^(n-1) / (2^((n-1)/2) * exp(n/2)). - Vaclav Kotesovec, Aug 05 2025

cp's OEIS Frontend

A362523 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (k! * (n-3*k)!).

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A361916 a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * (k+1)^(k-1) / (k! * (n-2*k)!).

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Formula

A362524 a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) / (2^k * k! * (n-2*k)!).

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