A202477 -id:A202477 - cp's OEIS frontend

A332236 E.g.f.: -log(2 - 1 / (1 + LambertW(-x))).

1, 5, 41, 466, 6769, 119736, 2497585, 60037328, 1634619969, 49733223040, 1672657257721, 61636181886720, 2470033974057649, 106970912288285696, 4979259164362745025, 247940951411958163456, 13152705012933836446465, 740578125097986605678592, 44115815578591964641401289

Offset: 1

Ilya Gutkovskiy, Feb 07 2020

nonn

Cf. A000312, A001865, A133297, A202477, A210661, A308863, A323673, A332237.

nmax = 19; CoefficientList[Series[-Log[2 - 1/(1 + LambertW[-x])], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = n^n + (1/n) Sum[Binomial[n, k] (n - k)^(n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]

E.g.f.: -log(1 - Sum_{k>=1} k^k * x^k / k!).
a(n) = n^n + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k) * k * a(k).
a(n) ~ (n-1)! * 2^n * exp(n/2). - Vaclav Kotesovec, Feb 16 2020

A352843 Expansion of e.g.f. exp(Sum_{k>=1} sigma_k(k) * x^k/k!).

Original entry on oeis.org

1, 1, 6, 44, 491, 6597, 110652, 2144606, 47988524, 1206275925, 33777572464, 1040200674416, 34967153135940, 1273241146218823, 49928549099500206, 2097300313258417056, 93953420539864844743, 4470694981375022862697, 225184078001798318202935

Offset: 0

Seiichi Manyama, Apr 05 2022

nonn

Exponential transform of A023887.

Cf. A295739, A274804, A352694.

Cf. A023887, A352841, A352842, A202477.

nmax = 20; CoefficientList[Series[E^(Sum[DivisorSigma[k, k]*x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sigma(k, k)*x^k/k!))))

a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*binomial(n-1, k-1)*a(n-k)));

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

cp's OEIS Frontend

A332236 E.g.f.: -log(2 - 1 / (1 + LambertW(-x))).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula