A362493 -id:A362493 - cp's OEIS frontend

A362492 E.g.f. satisfies A(x) = exp(x - x^2/2 * A(x)^2).

1, 1, 0, -8, -38, 106, 3676, 24508, -296036, -9149156, -56500064, 2211573376, 64958496472, 184823374360, -35372361487280, -971135892546224, 4364710018963216, 1034808592156017424, 25290798052846014208, -474242641154857953152, -49625273567646267051104

Offset: 0

Seiichi Manyama, Apr 22 2023

sign

Robert Israel, Table of n, a(n) for n = 0..403
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362493, A362494.

Cf. A362480.

N:= 50: # for a(0)..a(N)
egf:=  exp(x - LambertW(x^2 * exp(2*x))/2):
S:=series(egf,x,N+1):
[seq](coeff(S,x,i)*i!,i=0..N); # Robert Israel, May 22 2023

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2*exp(2*x))/2)))

E.g.f.: exp(x - LambertW(x^2 * exp(2*x))/2) = sqrt( LambertW(x^2 * exp(2*x))/x^2 ).

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

A362494 E.g.f. satisfies A(x) = exp(x - x^4/4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, -5, -149, -2249, -26249, -251159, -1443959, 21646801, 1209344401, 35457894451, 817789456771, 14796993881671, 137893562065351, -4661597156689199, -372730180154530799, -16419790692323174879, -559989133713039523679, -14492546886670841884949

Offset: 0

Seiichi Manyama, Apr 22 2023

sign

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A362492, A362493.

Cf. A362491.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^4*exp(4*x))/4)))

E.g.f.: exp(x - LambertW(x^4 * exp(4*x))/4) = ( LambertW(x^4 * exp(4*x))/x^4 )^(1/4).

a(n) = n! * Sum_{k=0..floor(n/4)} (-1/4)^k * (4*k+1)^(n-3*k-1) / (k! * (n-4*k)!).

cp's OEIS Frontend

A362492 E.g.f. satisfies A(x) = exp(x - x^2/2 * A(x)^2).

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