A354137 -id:A354137 - cp's OEIS frontend

A354136 Expansion of e.g.f. exp(log(1 + x)^3/6).

1, 0, 0, 1, -6, 35, -215, 1414, -9912, 73044, -552570, 4102626, -26654826, 79506492, 2154425364, -73527421176, 1708053626880, -35961691589640, 736338276883080, -15067241745943680, 312009998091705720, -6579362641255341120, 141704946709227843480

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A354137.

Cf. A327504, A347002, A354134.

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

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

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/(6^k*k!));

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

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/(6^k * 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!.

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A354136 Expansion of e.g.f. exp(log(1 + x)^3/6).

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

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