A356460 -id:A356460 - cp's OEIS frontend

A356461 Expansion of e.g.f. ( Product_{k>0} B(x^k)^k )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

1, 1, 8, 62, 631, 7417, 104489, 1648845, 29319588, 572982162, 12250559615, 283321630605, 7053444523393, 187711377451249, 5317981377046420, 159652557864884330, 5061465465801168419, 168886786171198864725, 5914650120884760212977, 216844308186908733542877

Offset: 0

Seiichi Manyama, Aug 08 2022

nonn

Cf. A356025, A356458.

Cf. A000110, A356459, A356460.

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)^(1/(1-x))))

a356459(n) = n!*sum(k=1, n, sumdiv(k, d, 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, a356459(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A356459(k) * binomial(n-1,k-1) * a(n-k).

A356494 Expansion of e.g.f. Product_{k>0} B(k * x^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

Original entry on oeis.org

1, 1, 6, 35, 327, 2892, 37943, 459895, 7330172, 116054835, 2168292295, 41072348550, 898738816957, 19782331776937, 487091519709590, 12305361661242275, 337777113607935587, 9528258228302443724, 289373132780801591323, 9016757353084706862647

Offset: 0

Seiichi Manyama, Aug 09 2022

nonn

Cf. A209903, A346055, A356495.

Cf. A000110, A354843, A356460.

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(k*x^k)-1))))

a354843(n) = n!*sumdiv(n, d, (n/d)^d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354843(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A354843(k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A356461 Expansion of e.g.f. ( Product_{k>0} B(x^k)^k )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

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A356494 Expansion of e.g.f. Product_{k>0} B(k * x^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

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Programs

PARI

PARI

Formula