A354136 -id:A354136 - cp's OEIS frontend

A354137 Expansion of e.g.f. exp(log(1 + x)^5/120).

1, 0, 0, 0, 0, 1, -15, 175, -1960, 22449, -269199, 3410000, -45753180, 650179816, -9771920158, 155020385156, -2589888417480, 45461879164584, -836540418765834, 16099972965770778, -323385447259166454, 6764948641797695496, -147088325599708573080

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A354136.

Cf. A327506, A347004, A354135.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5/120)))

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, 5, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/(120^k*k!));

a(0) = 1; a(n) = 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)/(120^k * k!).

A354231 Expansion of e.g.f. exp(log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -990, 2184, 37128, -863736, 13020480, -168384744, 1940801544, -18825129648, 107706637584, 1386022834944, -73429347222720, 2034345021802560, -46869707752067520, 976421492688165120, -18675350766042871680, 319467427583225518080

Offset: 0

Seiichi Manyama, May 20 2022

sign

Cf. A009199, A354232.

Cf. A353344, A354136, A354229.

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

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

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-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)/k!);

E.g.f.: (1 + x)^(log(1 + x)^2).
a(0) = 1; a(n) = 6 * 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)/k!.

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A354137 Expansion of e.g.f. exp(log(1 + x)^5/120).

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A354231 Expansion of e.g.f. exp(log(1 + x)^3).

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