A362796 -id:A362796 - cp's OEIS frontend

A362794 E.g.f. satisfies A(x) = (1+x)^(A(x)^x).

1, 1, 0, 6, 0, 170, -120, 12446, -35336, 1832400, -12172320, 469680552, -5524990416, 189586178184, -3321122831208, 111608536026360, -2599887499382400, 90253048158627072, -2595580675897337856, 95720854442948910720, -3237436187047116892800

Offset: 0

Seiichi Manyama, May 04 2023

sign

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

Cf. A033917, A362795.
Cf. A362796, A362799.
Cf. A349504, A349505.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^exp(-lambertw(-x*log(1+x)))))

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

E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (log(1+x))^k / k!.

A362798 E.g.f. satisfies A(x) = 1/(1-x)^(A(x)^(x^2)).

Original entry on oeis.org

1, 1, 2, 6, 48, 360, 2820, 31500, 393568, 5111568, 78491520, 1345893120, 24286008384, 483716087712, 10526811186528, 241867328844960, 5957816820215040, 157412355684364800, 4380674530640290560, 128826276098289179904, 4010282529115722232320

Offset: 0

Seiichi Manyama, May 04 2023

nonn

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

Cf. A052813, A362796.

Cf. A362795, A362800.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(-lambertw(x^2*log(1-x)))))

E.g.f.: exp( -LambertW(x^2 * log(1-x)) / x^2 ) = 1/(1-x)^exp( -LambertW(x^2 * log(1-x)) ).

E.g.f.: Sum_{k>=0} (k*x^2 + 1)^(k-1) * (-log(1-x))^k / k!.

A362799 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^x ).

Original entry on oeis.org

1, 1, 2, 11, 63, 542, 5183, 62211, 830252, 12900381, 220566835, 4223662522, 88001471869, 2007052809465, 49309469989666, 1306455781607975, 36973887007453315, 1116728635342926570, 35775769695237122035, 1213704083311914974899

Offset: 0

Seiichi Manyama, May 04 2023

nonn

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

Cf. A052880, A362800.
Cf. A362794, A362796.
Cf. A000110, A361777.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(exp(x)-1))/x)))

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

E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (exp(x) - 1)^k / k!.

cp's OEIS Frontend

A362794 E.g.f. satisfies A(x) = (1+x)^(A(x)^x).

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A362798 E.g.f. satisfies A(x) = 1/(1-x)^(A(x)^(x^2)).

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A362799 E.g.f. satisfies A(x) = exp( (exp(x) - 1) * A(x)^x ).

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Formula