A362472 -id:A362472 - 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

A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 25, 601, 9001, 105001, 1231441, 24146641, 740098801, 22443260401, 607394284201, 16102368745321, 497289446373721, 19072987370400601, 806135144596672801, 33945128330918599201, 1426006261391514829921, 63478993000497055809121

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

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

Cf. A143768, A349562, A362472.

Cf. A362393, A362482, A362491.

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)} (4*k+1)^(n-3*k-1) / (k! * (n-4*k)!).

A362481 E.g.f. satisfies A(x) = exp(x - x^3 * A(x)^3).

Original entry on oeis.org

1, 1, 1, -5, -95, -959, -5159, 69721, 3113377, 64493857, 654012721, -13761498959, -1013114081759, -32273321679455, -492845589685175, 13357113599586121, 1410278045569310401, 61239473424756703681, 1270682827211021594977, -40402942687262364034463

Offset: 0

Seiichi Manyama, Apr 21 2023

sign

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

Cf. A362480, A362482.

Cf. A362430, A362472.

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)} (-1)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

cp's OEIS Frontend

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

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A362473 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)^4).

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A362481 E.g.f. satisfies A(x) = exp(x - x^3 * A(x)^3).

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