%I #35 Feb 08 2023 07:24:21

%S 1,1,5,37,393,5481,95053,1975821,47939601,1330923601,41629292181,

%T 1448989481589,55561575788953,2327512861252281,105767732851318749,

%U 5182512561142513501,272391086209524010017,15287595381259195453089,912525533175190887597349,57726267762799335649572549

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

%H G. C. Greubel, <a href="/A273954/b273954.txt">Table of n, a(n) for n = 0..370</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: -LambertW(-x*exp(x)) / (x*exp(x)). [corrected by _Vaclav Kotesovec_, Jun 23 2016]

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

%F a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n-1) * LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jun 23 2016

%F E.g.f.: A(x) = exp(x*exp(x)*A(x)). - _Alexander Burstein_, Aug 11 2018

%F From _Peter Luschny_, Jan 29 2023: (Start)

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

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

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

%e 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! +...

%e such that

%e 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 +...

%e The logarithm of A(x) begins:

%e 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! +...

%e which equals -LambertW(-x*exp(x)).

%p A273954 := n -> add(binomial(n, j) * j^(n - j) * (j + 1)^(j - 1), j = 0..n):

%p seq(A273954(n), n = 0..24); # _Peter Luschny_, Jan 29 2023

%t CoefficientList[Series[-LambertW[-x*E^x] / (x*E^x), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 23 2016 *)

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) x='x+O('x^50); Vec(serlaplace(-lambertw(-x*exp(x))/(x*exp(x)))) \\ _G. C. Greubel_, Nov 16 2017

%o (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

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

%Y Cf. A161628, A244119.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 14 2016