A362351 -id:A362351 - cp's OEIS frontend

A362350 a(n) = n! * Sum_{k=0..floor(n/2)} (k/2)^k / (k! * (n-2*k)!).

1, 1, 2, 4, 19, 71, 601, 3277, 39089, 277489, 4250341, 37110701, 693581197, 7184750509, 158461520309, 1899055549861, 48269252293201, 656869268651537, 18903165795857089, 287927838327392929, 9252988524143245181, 155954097639111859501

Offset: 0

Seiichi Manyama, Apr 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..437
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362351, A362352.

Cf. A277614.

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

E.g.f.: exp(x) / (1 + LambertW(-x^2/2)).

a(n) ~ (exp(2^(3/2)*exp(-1/2)) + (-1)^n) * n^n / (2^((n+1)/2) * exp(n/2 + sqrt(2)*exp(-1/2))). - 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

A362705 Expansion of e.g.f. 1/(1 + LambertW(-x^3/6 * exp(x))).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 60, 595, 4536, 34524, 361320, 4333725, 51214460, 651628406, 9448719644, 146868322055, 2376666773040, 41077757951000, 762599081332176, 14918668387075449, 305774990501285940, 6602482711971622210, 149921553418087172260, 3557552268845721893131

Offset: 0

Seiichi Manyama, Apr 30 2023

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

Cf. A072034, A362704.

Cf. A362351.

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

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-2*k) / (6^k * k! * (n-3*k)!).

A362525 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (6^k * k! * (n-3*k)!).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 51, 246, 897, 7085, 51221, 260426, 2938541, 28279967, 184234415, 2714662406, 32614422401, 259026339161, 4721237878537, 67998862785970, 637019875964341, 13852253151455251, 232584488748665131, 2510358957337412182, 63466995535914172225

Offset: 0

Seiichi Manyama, Apr 23 2023

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

Cf. A088957, A362524.

Cf. A362351, A362523.

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

E.g.f.: exp(x - LambertW(-x^3/6)) = -6 * LambertW(-x^3/6)/x^3 * exp(x).

cp's OEIS Frontend

A362350 a(n) = n! * Sum_{k=0..floor(n/2)} (k/2)^k / (k! * (n-2*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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A362705 Expansion of e.g.f. 1/(1 + LambertW(-x^3/6 * exp(x))).

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A362525 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) / (6^k * k! * (n-3*k)!).

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