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
Keywords
Examples
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)).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..370
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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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
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Mathematica
CoefficientList[Series[-LambertW[-x*E^x] / (x*E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 23 2016 *) -
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
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)