A362494 -id:A362494 - cp's OEIS frontend

A362491 E.g.f. satisfies A(x) = exp(x + x^4/4 * A(x)^4).

1, 1, 1, 1, 7, 151, 2251, 26251, 273841, 3281041, 61021801, 1518719401, 38199828151, 905801252071, 21398411003971, 560160675014851, 17260034904184801, 596005144436100001, 21359751419836426321, 773082506262449261521, 28839945213850125032551

Offset: 0

Seiichi Manyama, Apr 22 2023

nonn

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

Cf. A362474, A362478.

Cf. A362473, A362494.

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)!).

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

Original entry on oeis.org

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)!).

A362493 E.g.f. satisfies A(x) = exp(x - x^3/3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, -1, -31, -319, -2279, -4199, 269473, 7155233, 114846641, 920526641, -18415853279, -1115017249631, -31675298017271, -526379460621559, 2394778195929281, 603748739138745281, 27895091311964499553, 769764386129113157473, 6164705700089328588481

Offset: 0

Seiichi Manyama, Apr 22 2023

sign

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

Cf. A362492, A362494.

Cf. A362478.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^3*exp(3*x))/3)))

E.g.f.: exp(x - LambertW(x^3 * exp(3*x))/3) = ( LambertW(x^3 * exp(3*x))/x^3 )^(1/3).

a(n) = n! * Sum_{k=0..floor(n/3)} (-1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

cp's OEIS Frontend

A362491 E.g.f. satisfies A(x) = exp(x + x^4/4 * A(x)^4).

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

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A362493 E.g.f. satisfies A(x) = exp(x - x^3/3 * A(x)^3).

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PARI

Formula