A361557 -id:A361557 - cp's OEIS frontend

A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

1, 1, 4, 19, 118, 886, 7786, 78184, 881644, 11017108, 150966856, 2249261356, 36181351504, 624658612384, 11516406883528, 225740649754936, 4686671645814736, 102712289940757264, 2369128149877075264, 57359541280704038128, 1454229915957292684576

Offset: 0

Seiichi Manyama, Mar 15 2023

nonn

Cf. A361548, A361557, A361558.

Cf. A052845, A112242, A185369.

With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 08 2023 *)

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

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - binomial(n-1,2) * a(n-3) for n > 2.

a(n) ~ 2^(-3/4) * 3^(1/4) * exp(-5/4 + sqrt(6*n) - n) * n^(n - 1/4) * (1 + sqrt(3)/(2*sqrt(2*n))). - Vaclav Kotesovec, Mar 20 2023

A361548 Expansion of e.g.f. exp((x + x^2/2 + x^3/6)/(1-x)).

Original entry on oeis.org

1, 1, 4, 20, 126, 966, 8656, 88544, 1016380, 12920156, 179996816, 2725070096, 44521522024, 780344770440, 14599772973696, 290311643773376, 6112190642062096, 135798496839920144, 3174483084427144000, 77872118431269176896, 1999809157085214044896

Offset: 0

Seiichi Manyama, Mar 15 2023

nonn

Cf. A361532, A361557, A361558.

With[{nn=20},CoefficientList[Series[Exp[(x+x^2/2+x^3/6)/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2+x^3/6)/(1-x))))

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - 2*binomial(n-1,3) * a(n-4) for n > 3.

A361558 Expansion of e.g.f. exp((x + x^2/2 + x^3/6 + x^4/24)/(1-x)).

Original entry on oeis.org

1, 1, 4, 20, 127, 976, 8776, 90084, 1037555, 13233077, 184956386, 2809098986, 46038214729, 809411443790, 15189361799522, 302932433571356, 6396529241755881, 142523960797017589, 3341115707515530400, 82187749261419720712, 2116421112495023612311

Offset: 0

Seiichi Manyama, Mar 15 2023

nonn

Cf. A361532, A361548, A361557.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x+x^2/2+x^3/6+x^4/24)/(1-x))))

a(n) = (2*n-1) * a(n-1) - (n-1)*(n-3) * a(n-2) - binomial(n-1,3) * a(n-4) - 3*binomial(n-1,4) * a(n-5) for n > 4.

cp's OEIS Frontend

A361532 Expansion of e.g.f. exp((x + x^2/2)/(1-x)).

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A361548 Expansion of e.g.f. exp((x + x^2/2 + x^3/6)/(1-x)).

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A361558 Expansion of e.g.f. exp((x + x^2/2 + x^3/6 + x^4/24)/(1-x)).

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Keywords

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PARI

Formula