A360547 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^2 ).

1, 1, 9, 121, 2417, 64721, 2180665, 88719625, 4233968737, 231991022881, 14356691152361, 990506937621785, 75390334060230865, 6275675303410022641, 567191776288882702105, 55313848534122299876521, 5789703106014903009828545, 647414950001156861671249985

Offset: 0

Author

Seiichi Manyama, Feb 11 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..339
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A273953, A273954, A360544.

Cf. A360465, A360473.

Programs

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 *)

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));

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