A362473 -id:A362473 - cp's OEIS frontend

A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

1, 1, 1, 7, 97, 961, 10201, 177241, 3801505, 80718625, 1887205681, 52896262321, 1648697978401, 54216677033377, 1928791931034697, 75326014326206281, 3159713152034201281, 140373558362282197441, 6632746205445950124385, 333591744669464008432225

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..384
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362490.
Cf. A143768, A349562, A362473.
Cf. A362392, A362481.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^3*exp(3*x))/3)))

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

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

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

Original entry on oeis.org

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)!).

A362482 E.g.f. satisfies A(x) = exp(x - x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, -23, -599, -8999, -104999, -868559, 5246641, 582598801, 21205760401, 571129277401, 11475082596121, 81837031796521, -7904119577117399, -596529385424263199, -28051840646006771999, -991870986521074646879, -21837506791918601443679

Offset: 0

Seiichi Manyama, Apr 21 2023

sign

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

Cf. A362480, A362481.

Cf. A362431, A362473.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(4*x^4*exp(4*x))/4)))

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

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

cp's OEIS Frontend

A362472 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)^3).

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

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A362482 E.g.f. satisfies A(x) = exp(x - x^4 * A(x)^4).

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