OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..339
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
E.g.f.: A(x) = exp( (-1/2) * LambertW(-2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(2*x)) / (2*x * exp(2*x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(2*x))^k / k! ).
a(n) = Sum_{k=0..n} (2*k)^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (exp(n - 1/2) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x]*A[x])^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*exp(2*x))/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-lambertw(-2*x*exp(2*x))/(2*x*exp(2*x)))))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(2*x))^k/k!))))
(PARI) a(n) = sum(k=0, n, (2*k)^(n-k)*(2*k+1)^(k-1)*binomial(n, k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 11 2023
STATUS
approved