A052888 -id:A052888 - cp's OEIS frontend

A334258 a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.

1, 1, 1, -5, -74, -679, -4899, -17289, 325837, 10627109, 199348590, 2684041427, 15872610469, -546948563407, -27499774835519, -778467357484561, -15311413773551790, -125363405319188419, 6452292137017871097, 436442148982835915339, 16494863323310244977581

Offset: 1

Ilya Gutkovskiy, Apr 20 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A002741, A052888, A292866.

nmax = 21; CoefficientList[InverseSeries[Series[-Log[1 - x] Exp[-x], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest
Table[Sum[(-1)^(n - k) StirlingS2[n, k] n^(k - 1), {k, 1, n}], {n, 1, 21}]
Table[(-1)^n BellB[n, -n]/n, {n, 1, 21}]

a(n) = sum(k=1, n, (-1)^(n-k) * stirling(n,k,2) * n^(k-1)); \\ Michel Marcus, Apr 20 2020

E.g.f.: series reversion of -log(1 - x) * exp(-x).
a(n) = (n - 1)! * [x^n] exp(n*(1 - exp(-x))).
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling2(n,k) * n^(k-1).
a(n) = (-1)^n * BellPolynomial_n(-n) / n.

A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

Original entry on oeis.org

1, 2, 8, 55, 507, 5969, 85605, 1445420, 28110615, 618760615, 15207174501, 412790636977, 12265337498000, 395962288858946, 13800754780797740, 516494067220932259, 20658199248901273576, 879406216174705907137, 39698270229941320201019, 1894212537494300993244732

Offset: 1

Paul Laubie, Nov 29 2023

nonn

See A367752 for the definition of the shape of a labeled rooted hypertree.

Cf. A052888, A210586, A364709, A364816, A367752.

my(x='x+O('x^30)); Vec(serlaplace(serreverse((1+x)*log(1+log(1+x))*exp(-x)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ,30);(1+t)*(ln(1+ln(1+t))*exp(-t)).reverse().egf_to_ogf().list()[1:]

E.g.f.: series reversion of (1+x)*log(1+log(1+x))*exp(-x).

cp's OEIS Frontend

A334258 a(n) = (-1)^n * exp(n) * Sum_{k>=1} (-1)^k * n^(k-1) * k^n / k!.

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