A273954 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * exp(n*x) * A(x)^n.

1, 1, 5, 37, 393, 5481, 95053, 1975821, 47939601, 1330923601, 41629292181, 1448989481589, 55561575788953, 2327512861252281, 105767732851318749, 5182512561142513501, 272391086209524010017, 15287595381259195453089, 912525533175190887597349, 57726267762799335649572549

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..370

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

E.g.f.: -LambertW(-x*exp(x)) / (x*exp(x)). [corrected by Vaclav Kotesovec, Jun 23 2016]

E.g.f.: exp( L(x) ) where L(x) = -LambertW(-x*exp(x)) is the e.g.f. of A216857.

a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016

E.g.f.: A(x) = exp(x*exp(x)*A(x)). - Alexander Burstein, Aug 11 2018

From Peter Luschny, Jan 29 2023: (Start)

a(n) = Sum_{j=0..n} binomial(n, j) * j^(n - j) * (j + 1)^(j - 1).

a(n) = Sum_{k=0..n} (-1)^k*A161628(n, k).

a(n) = Sum_{k=0..n} (-1)^(n-k)*A244119(n, k). (End)

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 393*x^4/4! + 5481*x^5/5! + 95053*x^6/6! + 1975821*x^7/7! + 47939601*x^8/8! + 1330923601*x^9/9! + 41629292181*x^10/10! + 1448989481589*x^11/11! + 55561575788953*x^12/12! +...
such that
A(x) = 1 + x*exp(x)*A(x) + x^2/2!*exp(2*x)*A(x)^2 + x^3/3!*exp(3*x)*A(x)^3 + x^4/4!*exp(4*x)*A(x)^4 + x^5/5!*exp(5*x)*A(x)^5 + x^6/6!*exp(6*x)*A(x)^6 +...
The logarithm of A(x) begins:
log(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! + 600698880*x^9/9! + 18422374400*x^10/10! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Maple

A273954 := n -> add(binomial(n, j) * j^(n - j) * (j + 1)^(j - 1), j = 0..n):
seq(A273954(n), n = 0..24); # Peter Luschny, Jan 29 2023

Mathematica

CoefficientList[Series[-LambertW[-x*E^x] / (x*E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 23 2016 *)

Prog

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m!*exp(m*x +x*O(x^n))*A^m) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) x='x+O('x^50); Vec(serlaplace(-lambertw(-x*exp(x))/(x*exp(x)))) \\ G. C. Greubel, Nov 16 2017
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x*exp(x))^k/k!))) \\ Seiichi Manyama, Feb 08 2023

Crossrefs

Cf. A273953, A216857, A357247, A360176 (column 1 unsigned).

Cf. A161628, A244119.

Sequence in context: A112937 A258378 A368322 * A092649 A179923 A190628

Adjacent sequences: A273951 A273952 A273953 * A273955 A273956 A273957

Keyword

nonn

Author

Paul D. Hanna, Jun 14 2016

Status

approved