A360483 -id:A360483 - cp's OEIS frontend

A360481 E.g.f. satisfies A(x) = x * exp(x + 2 * A(x)).

0, 1, 6, 63, 1044, 23805, 692118, 24482115, 1020584232, 49000005945, 2662853279850, 161586078510879, 10830019921469532, 794577001293803637, 63339899145968483262, 5451312770064188283195, 503784284643602483767632, 49757423537114340032969073

Offset: 0

Seiichi Manyama, Feb 09 2023

nonn

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

Cf. A216857, A360482, A360483, A360484.

Cf. A360473.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(x))/2)))

a(n) = sum(k=1, n, 2^(k-1)*k^(n-1)*binomial(n, k));

E.g.f.: A(x) = -LambertW(-2*x * exp(x))/2.
a(n) = Sum_{k=1..n} 2^(k-1) * k^(n-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * LambertW(exp(-1)/2)^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

A360482 E.g.f. satisfies A(x) = x * exp(x + 3 * A(x)).

Original entry on oeis.org

0, 1, 8, 120, 2848, 92960, 3868224, 195810496, 11680512512, 802445898240, 62396469222400, 5417515922441216, 519519435065020416, 54535504354085687296, 6219954774471102242816, 765903524713482618101760, 101269330068289021683564544, 14310318526812295078276628480

Offset: 0

Seiichi Manyama, Feb 09 2023

nonn

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

Cf. A216857, A360481, A360483, A360484.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-3*x*exp(x))/3)))

a(n) = sum(k=1, n, 3^(k-1)*k^(n-1)*binomial(n, k));

E.g.f.: A(x) = -LambertW(-3*x * exp(x))/3.
a(n) = Sum_{k=1..n} 3^(k-1) * k^(n-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1)/3)) * n^(n-1) /(3 * exp(n) * LambertW(exp(-1)/3)^n). - Vaclav Kotesovec, Feb 17 2023

A360484 E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).

Original entry on oeis.org

0, 1, -4, 48, -896, 22880, -743232, 29337280, -1363752448, 72979407360, -4419108684800, 298730433250304, -22300928914403328, 1822195561572585472, -161756111552270491648, 15501595224386724126720, -1595092357302221461127168, 175405731698165304882495488

Offset: 0

Seiichi Manyama, Feb 09 2023

sign

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

Cf. A216857, A360481, A360482, A360483.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(3*x*exp(x))/3)))

a(n) = sum(k=1, n, (-3)^(k-1)*k^(n-1)*binomial(n, k));

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

a(n) = Sum_{k=1..n} (-3)^(k-1) * k^(n-1) * binomial(n,k).

cp's OEIS Frontend

A360481 E.g.f. satisfies A(x) = x * exp(x + 2 * A(x)).

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A360482 E.g.f. satisfies A(x) = x * exp(x + 3 * A(x)).

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A360484 E.g.f. satisfies A(x) = x * exp(x - 3 * A(x)).

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