A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

1, 1, 7, 85, 1521, 36421, 1097743, 39968601, 1707558401, 83777885929, 4643185678551, 286930307457949, 19562851003118833, 1458832806486727725, 118121195050068075167, 10320576944751955718881, 967863775658734350214017, 96970880819175875321264209

Offset: 0

Author

Seiichi Manyama, Mar 01 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..342
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A052868, A361066, A361067, A361068, A361069.

Programs

Maple

A361065 := proc(n)
    add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
    %*n! ;
end proc:
seq(A361065(n),n=0..10) ; # R. J. Mathar, Mar 02 2023

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x))/2)))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-(1-x)/(2*x)*lambertw(-2*x/(1-x)))))

Formula

a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n-1,n-k)/k!.

E.g.f.: exp( -LambertW(-2*x/(1-x))/2 ).

E.g.f.: sqrt( -(1-x)/(2*x) * LambertW(-2*x/(1-x)) ).

a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (2^(3/2) * exp(n)). - Vaclav Kotesovec, Mar 02 2023