A355337 -id:A355337 - cp's OEIS frontend

A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

1, 1, 0, -1, 3, 12, -7, -47, 332, 1347, -2105, -4200, 135457, 474697, -900832, 4682135, 126196787, 439488524, 233313817, 19129265609, 239146712732, 1104038984091, 5891696027079, 89831511761320, 911995655018817, 6253185308181553, 54873149768926624, 653039078246798383

Offset: 0

Vaclav Kotesovec, Jun 29 2022

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Vaclav Kotesovec, Table of n, a(n) for n = 0..578

Cf. A000296, A277381, A316778, A355337.

nmax = 30; CoefficientList[Series[Exp[Exp[x] - x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) - x^2 - 1))) \\ Michel Marcus, Jun 29 2022

a(n) ~ n^n * exp(n/LambertW(n) - LambertW(n)^2 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n).
a(n) ~ Bell(n) / exp(LambertW(n)^2).
a(0) = a(1) = 1; a(n) = -2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022

A360991 Expansion of e.g.f. exp(exp(x) - 1 + x^2/2).

Original entry on oeis.org

1, 1, 3, 8, 30, 117, 533, 2599, 13919, 79620, 487810, 3167265, 21744187, 157020697, 1189321019, 9417789650, 77774264012, 668233623419, 5961395449795, 55117233908411, 527263186773227, 5210880621612366, 53130216638022540, 558176360466846439

Offset: 0

Seiichi Manyama, Mar 14 2023

Cf. A097514, A355337, A361489.

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

a(0) = a(1) = 1; a(n) = (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

A361493 Expansion of e.g.f. exp(exp(x) - 1 + x^3).

Original entry on oeis.org

1, 1, 2, 11, 39, 172, 1163, 6547, 41772, 335139, 2486215, 20078610, 186139957, 1676540257, 16077206122, 168739976555, 1763716943267, 19358116589964, 226362412711759, 2669223655597955, 32748447519013132, 421204995451111971, 5496921281576148363

Offset: 0

Seiichi Manyama, Mar 14 2023

Cf. A355337, A361489.

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

a(0) = a(1) = 1, a(2) = 2; a(n) = 6 * binomial(n-1,2) * a(n-3) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

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A360991 Expansion of e.g.f. exp(exp(x) - 1 + x^2/2).

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A361493 Expansion of e.g.f. exp(exp(x) - 1 + x^3).

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