A277759 a(n) equals the coefficient of x^n in n!*(1 - log(1-x))^n.

1, 1, 4, 30, 324, 4540, 78060, 1589448, 37388400, 997513200, 29759790240, 981669324240, 35475203063520, 1393746645107232, 59147129937893088, 2696314664384853120, 131405475202661963520, 6817779852438948837120, 375193156508083422581760

Offset

0,3

Comments

For n>0, a(n) is the number of partitions of [n] that place one node into a box and the rest into an ordered forest of mobiles, where singleton mobiles (whose root is a leaf) can be painted. - Natalia L. Skirrow, Apr 20 2026

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..375

Natalia L. Skirrow, counting endofunctions on [n] with j trees and k leaves: interpreting a Lagrange inversion result, Mathematics StackExchange (2026); derivation of H(x,y) here is analogous to equation (*) there.

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * k! * (-1)^(n-k) * Stirling1(n,k).

a(n) ~ d^n * n^n / (sqrt(d-1) * exp(n)), where d = A226572 = -LambertW(-1, -exp(-2)) = 3.146193220620582585237...

From Natalia L. Skirrow, Apr 20 2026: (Start)

Let

* F(x) = LambertW(-1,-exp(1-1/x)/x),

* G(x) = e.g.f. of A138013,

* H(x) = e.g.f. of A038037,

then F(x) = G(x)-1/x and G(x) = H(x)/x.

E.g.f.: F(x)/(1 + F(x)) = 1 + x/(1 - x - H(x)).

Let H(x,y) = 1 + x*LambertW(-1,-exp(y-1/x)/x) be the e.g.f. of A055349 (such that H(x) = H(x,1)).

Then [x^n/n! * y^k] H(x,y)^r/r! = C(n-1,r-1) * n!/k! * |Stirling1(n-r,n-k)| is the number of forests of mobiles with n labelled nodes, of which r are roots and k are leaves.

E.g.f.: 1 + Sum_{r>=0} H(x)^r * x/(1-x)^(r+1), yielding:

a(n) = Sum_{r=0..n} r! * Sum_{i=r..n} C(n-i+r-1,r) * C(i-1,r-1) * Sum_{k=r..i} n!/k! * |Stirling1(i-r,i-k)|. (k in this sum (# leaves) corresponds to n-k in first sum, i is # mobiles, i-r is # painted singletons, r is # unpainted mobiles.)

(End)

Example

Denote the box as [], a mobile node as (), and a singleton painted the other way as {}.
a(1)=1: [1].
a(2)=4: [1](2), [1]{2}, [2](1), [2]{1}.
Hereafter singleton variants are omitted.
a(3)=30 = 6 + 24: [1](2:(3)) (*6 symbol permutations), [1](2)(3) (*6 symbol permutations, *4 paint combinations).
a(4)=324 = 24 + 12 + 2*48 + 192: [1](2:(3:(4))) (*4! perms), [1](2:(3)(4)) (*4!/2 perms since (3)(4) are cyclic), [1](2:(3))(4) and [1](4)(2:(3)) (*4! perms and *2 paintings each), [1](2)(3)(4) (*4! perms, *8 paint combs).

Mathematica

Table[n!*SeriesCoefficient[(1-Log[1-x])^n, {x, 0, n}], {n, 0, 20}]
Table[Sum[Binomial[n, k]*k!*(-1)^(n-k)*StirlingS1[n, k], {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A277409, A226572.

Sequence in context: A180623 A128329 A211828 * A392786 A387974 A006149

Adjacent sequences: A277756 A277757 A277758 * A277760 A277761 A277762

Keyword

nonn

Author

Vaclav Kotesovec, Oct 30 2016

Status

approved