A362350 -id:A362350 - cp's OEIS frontend

A362351 a(n) = n! * Sum_{k=0..floor(n/3)} (k/6)^k / (k! * (n-3*k)!).

1, 1, 1, 2, 5, 11, 61, 316, 1177, 11005, 84121, 434446, 5642781, 56725527, 374014005, 6211205456, 77331975281, 620174850521, 12539310726577, 186125334960730, 1757911008913141, 41887694462674691, 721886016954223661, 7846403629258814852, 215270385425700640905

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..469
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362350, A362352.

Cf. A362173.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(-x^3/6))))

E.g.f.: exp(x) / (1 + LambertW(-x^3/6)).

a(n) ~ n^n * (exp(6^(1/3)*exp(-1/3)) + 2*cos(2^(-2/3)*3^(5/6)*exp(-1/3) - 2*Pi*n/3) / exp(2^(-2/3)*3^(1/3)*exp(-1/3))) / (2^(n/3) * 3^(n/3 + 1/2) * exp(2*n/3)). - Vaclav Kotesovec, Apr 18 2023

A362352 a(n) = n! * Sum_{k=0..floor(n/4)} (k/24)^k / (k! * (n-4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 211, 1387, 6511, 23431, 225721, 2207921, 14610597, 71848141, 958259121, 12403693681, 105819536881, 659686502257, 11235532306021, 180826378073461, 1888306425160541, 14256573124903341, 295428115205647117, 5683724892725141901

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..494
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362350, A362351.

Cf. A362317.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)/(1+lambertw(-x^4/24))))

E.g.f.: exp(x) / (1 + LambertW(-x^4/24)).

a(n) ~ (exp(2^(3/4)*3^(1/4)*exp(-1/4)) + (-1)^n/exp(2^(3/4)*3^(1/4)*exp(-1/4)) + 2*cos(2^(3/4)*3^(1/4)*exp(-1/4) - Pi*n/2)) * n^n / (2^(3*n/4 + 1) * 3^(n/4) * exp(3*n/4)). - Vaclav Kotesovec, Apr 18 2023

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

A362704 Expansion of e.g.f. 1/(1 + LambertW(-x^2/2 * exp(x))).

Original entry on oeis.org

1, 0, 1, 3, 18, 130, 1140, 11886, 142408, 1934640, 29357100, 492249340, 9038206056, 180352513848, 3886286296984, 89937276717120, 2224716791224320, 58577968147130176, 1635780290409117648, 48286974141713673072, 1502385897082471446880

Offset: 0

Seiichi Manyama, Apr 30 2023

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

Cf. A072034, A362705.

Cf. A362350.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^2/2*exp(x)))))

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

cp's OEIS Frontend

A362351 a(n) = n! * Sum_{k=0..floor(n/3)} (k/6)^k / (k! * (n-3*k)!).

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A362352 a(n) = n! * Sum_{k=0..floor(n/4)} (k/24)^k / (k! * (n-4*k)!).

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

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A362704 Expansion of e.g.f. 1/(1 + LambertW(-x^2/2 * exp(x))).

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