A362393 -id:A362393 - cp's OEIS frontend

A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

1, 1, 1, 7, 49, 241, 2041, 26041, 282913, 3449377, 57170161, 973059121, 16847893921, 343341027745, 7680743819113, 175958943331081, 4375517632543681, 118932887426911681, 3374685950589927649, 100735118425384221025, 3217474234925998764481

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362378.

Cf. A118395, A125500, A362393, A362430.

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

A362431 E.g.f. satisfies A(x) = exp(x - x^4 * A(x)).

Original entry on oeis.org

1, 1, 1, 1, -23, -239, -1439, -6719, 33601, 1536193, 24171841, 268424641, 1144566721, -47515765439, -1727426116415, -36344982098879, -481057514071679, 1197767242412161, 319851095455612801, 12145632936380316289, 293167011107091899521, 3520557699737168603521

Offset: 0

Seiichi Manyama, Apr 20 2023

sign

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

Cf. A362396, A362430.

Cf. A362393.

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

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)!).

cp's OEIS Frontend

A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

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A362431 E.g.f. satisfies A(x) = exp(x - x^4 * A(x)).

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

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