A356401 -id:A356401 - cp's OEIS frontend

A356402 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^(1/(1-x)).

1, 1, 3, 16, 86, 626, 5267, 50793, 543279, 6544805, 86503762, 1242678141, 19259416827, 321457169151, 5736414618209, 108931865485750, 2191495621647324, 46604972526167314, 1043844453093239627, 24555321244430950299, 605239630722584461955, 15600222966916650541099

Offset: 0

Seiichi Manyama, Aug 05 2022

nonn

Cf. A298906, A356025, A356392, A356401.

my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k!))^(1/(1-x))))

a356401(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356401(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356401(k) * binomial(n-1,k-1) * a(n-k).

A354509 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(d+1)/(d * (k/d)!) )/(n-k)!.

Original entry on oeis.org

1, 2, 6, 5, 5, -8, 560, -5997, -14765, 176826, 5206410, -42491623, -427057527, -412183484, 147180377804, -569782989113, -8367671807033, -119681999820906, 4440973420854454, -121033449284728099, 49772248126885197, 36615485147317407728, 1696495197400394891912

Offset: 1

Seiichi Manyama, Aug 15 2022

sign

Cf. A352013, A354505, A356401.

a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!))/(n-k)!);

a352013(n) = sumdiv(n, d, (-1)^(n/d+1)*(n-1)!/(d-1)!);
a(n) = sum(k=1, n, a352013(k)*binomial(n, k));

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, log(1+x^k)/k!)))

a(n) = Sum_{k=1..n} A352013(k) * binomial(n,k).
E.g.f.: -exp(x) * Sum_{k>0} (-1)^k * (exp(x^k) - 1)/k.
E.g.f.: exp(x) * Sum_{k>0} log(1+x^k)/k!.

cp's OEIS Frontend

A356402 Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^(1/(1-x)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A354509 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(d+1)/(d * (k/d)!) )/(n-k)!.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

PARI

Formula