A377361 -id:A377361 - cp's OEIS frontend

A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

1, 2, 12, 130, 2082, 44488, 1192964, 38557860, 1459988440, 63414711072, 3108861424032, 169829819311392, 10230860299538400, 673850170929176928, 48176129912775680160, 3715759452364764485280, 307545698210584533055488, 27190399275422185989742080, 2557448587458299889542868480

Offset: 0

Seiichi Manyama, Oct 26 2024

Index entries for reversions of series

Cf. A138013, A377361.

Cf. A097629, A367080, A376392.

nmax = 20; CoefficientList[1/x * InverseSeries[Series[x/(1 - Log[1 - x])^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 27 2025 *)

a(n) = 2*(2*n+1)!*sum(k=0, n, abs(stirling(n, k, 1))/(2*n-k+2)!);

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367080.
a(n) = 2 * (2*n+1)! * Sum_{k=0..n} |Stirling1(n,k)|/(2*n-k+2)!.
E.g.f.: (1/x) * Series_Reversion( x/(1 - log(1-x))^2 ).
a(n) ~ sqrt(2) * LambertW(-1, -2*exp(-3))^n * (2 + LambertW(-1, -2*exp(-3)))^(n+2) * n^(n-1) / (exp(n) * sqrt(-1 - LambertW(-1, -2*exp(-3)))). - Vaclav Kotesovec, Aug 27 2025

A377391 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 - x*log(1-x))^3 ).

Original entry on oeis.org

1, 0, 6, 9, 528, 3150, 157032, 2060100, 102770112, 2276373456, 120136435200, 3868551141840, 221493499198848, 9438561453784320, 592954244405195904, 31417910131585330080, 2173884244961012121600, 137231093173511486016000, 10452538023125775799541760

Offset: 0

Seiichi Manyama, Oct 27 2024

nonn

Index entries for reversions of series

Cf. A371121, A377390.

Cf. A371231, A377361.

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

E.g.f. A(x) satisfies A(x) = ( 1 - x*A(x)*log(1 - x*A(x)) )^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A371231.
a(n) = 3 * n! * (3*n+2)! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/( (n-k)! * (3*n-k+3)! ).

cp's OEIS Frontend

A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula