A362392 -id:A362392 - cp's OEIS frontend

A362734 E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

1, 2, 16, 296, 8512, 333632, 16595200, 1001460224, 71094759424, 5805799829504, 536188352856064, 55259197654089728, 6287146625230962688, 782751635353947865088, 105852868748672770244608, 15451195442132410179780608, 2421355190097788960505856000

Offset: 0

Seiichi Manyama, May 01 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..322
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362693, A362694, A362735.

Cf. A349714, A362392, A362472.

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

E.g.f.: ( -LambertW(-3*x*exp(3*x)) / (3*x) )^(1/3) = exp( x - LambertW(-3*x*exp(3*x))/3 ).
a(n) = Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) = 2^n * A349714(n).
a(n) ~ sqrt(LambertW(exp(-1)) + 1) * 3^(n-1) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n + 1/3)). - Vaclav Kotesovec, Apr 24 2024

A362393 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)).

Original entry on oeis.org

1, 1, 1, 1, 25, 241, 1441, 6721, 87361, 1729729, 24816961, 270452161, 3705324481, 85344916801, 1992230175937, 38047293910081, 709217112938881, 17385498239168641, 514103858592923521, 14254662916125735553, 366807994235438359681, 10338786602768939575681

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..422
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A125500, A362392.

Cf. A362431.

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

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

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

A362430 E.g.f. satisfies A(x) = exp(x - x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, -5, -47, -239, 121, 19321, 261409, 1449505, -20428559, -730564559, -10403326559, -10910781023, 3713153976169, 108037345645321, 1301173754543041, -22441761904964159, -1628466860540690207, -41339196023230498463, -189173461196772118079

Offset: 0

Seiichi Manyama, Apr 20 2023

sign

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

Cf. A362396, A362431.

Cf. A362392.

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

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

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

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

Original entry on oeis.org

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)!).

A372250 E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x^2 * A(x)) ).

Original entry on oeis.org

1, 1, 3, 22, 221, 2796, 44527, 857074, 19250457, 494672824, 14331709691, 462277166814, 16430392693717, 638087135544340, 26885810437578471, 1221643263680551786, 59548603124180235953, 3099741728641090867056, 171619335190080336331891

Offset: 0

Seiichi Manyama, Apr 24 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..376

Cf. A088695, A138764.

Cf. A362392.

a(n) = n!*sum(k=0, n\3, (n-k+1)^(n-2*k-1)/(k!*(n-3*k)!));

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

A372234 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x^2 * A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 4, 20, 160, 1112, 9424, 114788, 1453792, 19242224, 309179104, 5533486268, 102733943536, 2105041949480, 47732237414320, 1139969559931028, 28924667996076736, 792458458301707232, 22984740550326524608, 699915806697250558316

Offset: 0

Seiichi Manyama, Apr 23 2024

nonn

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

Cf. A362392, A372156.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-2*lambertw(-x^3*exp(x)))))

a(n) = 2*n!*sum(k=0, n\3, (k+2)^(n-2*k-1)/(k!*(n-3*k)!));

E.g.f.: A(x) = exp( 2*x - 2*LambertW(-x^3 * exp(x)) ).
a(n) = 2 * n! * Sum_{k=0..floor(n/3)} (k+2)^(n-2*k-1) / (k! * (n-3*k)!).
a(n) ~ 2*sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (3^(n + 11/2) * exp(n) * LambertW(exp(-1/3)/3)^(n+6)). - Vaclav Kotesovec, Jun 01 2024

cp's OEIS Frontend

A362734 E.g.f. satisfies A(x) = exp(x + x * A(x)^3).

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A362393 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)).

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A362430 E.g.f. satisfies A(x) = exp(x - x^3 * A(x)).

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

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A372250 E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x^2 * A(x)) ).

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A372234 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x^2 * A(x)^(1/2)) ).

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