A354230 -id:A354230 - cp's OEIS frontend

A354229 Expansion of e.g.f. 1/(1 - log(1 + x)^3).

1, 0, 0, 6, -36, 210, -630, -5376, 153048, -2194296, 22190760, -93956544, -2677330656, 97821857952, -2019503487456, 27899293618944, -98409183995520, -9548919666829440, 410311098024923520, -10652005874894469120, 176525303194482117120, -46197517147757867520

Offset: 0

Seiichi Manyama, May 20 2022

sign

Cf. A006252, A354230.

Cf. A353118, A354134, A354231.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*stirling(j, 3, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1));

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * Stirling1(k,3) * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k).

A354232 Expansion of e.g.f. exp(log(1 + x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, -1800, 21000, -235200, 2693880, -30504600, 310239600, -2026767600, -22324267680, 1480359360480, -48314853350400, 1332965821824000, -34178451017685120, 837433109548661760, -19671723873906894720, 436228097513559408000

Offset: 0

Seiichi Manyama, May 20 2022

sign

Cf. A009199, A354231.

Cf. A354137, A354230.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5)))

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^4))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/k!);

E.g.f.: (1 + x)^(log(1 + x)^4).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/k!.

cp's OEIS Frontend

A354229 Expansion of e.g.f. 1/(1 - log(1 + x)^3).

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A354232 Expansion of e.g.f. exp(log(1 + x)^5).

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