A354339 -id:A354339 - cp's OEIS frontend

A354341 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * ((k/d)!)^d) )/(n-k)!.

1, 4, 12, 38, 130, 557, 2877, 18314, 136458, 1180457, 11389081, 122833207, 1446973931, 18594740348, 257507754524, 3835059283282, 60937544854850, 1030871972064485, 18469079943443229, 349656695460113159, 6969526853682012755, 145958486484692023936

Offset: 1

Seiichi Manyama, Aug 15 2022

nonn

Cf. A182926, A354339, A356407.

a182926(n) = n!*sumdiv(n, d, 1/(d*(n/d)!^d));
a(n) = sum(k=1, n, a182926(k)*binomial(n, 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} A182926(k) * binomial(n,k).

E.g.f.: -exp(x) * Sum_{k>0} log(1-x^k/k!).

A356597 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^exp(x).

Original entry on oeis.org

1, 1, 5, 26, 172, 1354, 12403, 127945, 1471006, 18589503, 255951308, 3808299648, 60871219649, 1039240205691, 18868377309780, 362838034712928, 7364831540699076, 157305165900364641, 3526069495916583260, 82744901973286823822, 2028396974232995349291

Offset: 0

Seiichi Manyama, Aug 15 2022

nonn

Cf. A354339, A356408.

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

a354339(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d))/(n-k)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354339(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A354339(k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A354341 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * ((k/d)!)^d) )/(n-k)!.

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A356597 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^exp(x).

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