A276911 E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x).

1, 2, 6, 28, 180, 1446, 13888, 156472, 2034000, 29724490, 476806176, 8502508884, 174802753216, 3768345692398, 63300353418240, 1386349221087856, 149879079531401472, 5097575010920072850, -780487993325688128000, -32524149870689487270260, 10927977097616993825596416, 490896441869732669067535414, -213936255246865273137807851520, -10450262329586550037066790750808, 6047981224337998054714885264691200

Offset

1,2

Comments

Former name was "Inverse of e.g.f. A(x) equals its conjugate, where A(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1)." - Paul D. Hanna, Sep 06 2018

Links

Paul D. Hanna, Table of n, a(n) for n = 1..301

Formula

E.g.f. A(x) satisfies: A(A( x*exp(-x) )) = x*exp(x). - Paul D. Hanna, Sep 06 2018

E.g.f. A(x) satisfies: A(-A(-x)) = x. - Paul D. Hanna, Sep 06 2018

Inverse of F(x) equals its conjugate, where F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! and i=sqrt(-1).

Let G(x) be the e.g.f. of A276910, then F(x) = Sum_{n>=1} a(n)*i^(n-1)*x^n/n! satisfies:

(1) F(x) = G(x) * exp(i*G(x)).

(2) G( F(x) ) = i*LambertW(-i*x), where LambertW( x*exp(x) ) = x.

E.g.f. A(x) satisfies: A(A(x)) is e.g.f. of A089946 with offset 1. - Alexander Burstein, Jan 15 2022

Example

E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + ...
such that A(A( x*exp(-x) )) = x*exp(x).
RELATED SERIES.
Let F(x) = x + 2*I*x^2/2! - 6*x^3/3! - 28*I*x^4/4! + 180*x^5/5! + 1446*I*x^6/6! - 13888*x^7/7! - 156472*I*x^8/8! + 2034000*x^9/9! + 29724490*I*x^10/10! - 476806176*x^11/11! - 8502508884*I*x^12/12! + 174802753216*x^13/13! + 3768345692398*I*x^14/14! - 63300353418240*x^15/15! - 1386349221087856*I*x^16/16! + 149879079531401472*x^17/17! +...+ a(n)*i^(n-1)*x^n/n! +...
then
(a) Series_Reversion( F(x) ) = conjugate( F(x) ).
(b) F(x) = G(x)*exp(i*G(x)) where G(x) is the e.g.f. of A276910:
(c) G(x) = x - 3*x^3/3! + 85*x^5/5! - 6111*x^7/7! + 872649*x^9/9! - 195062395*x^11/11! + 76208072733*x^13/13! - 12330526252695*x^15/15! + 125980697776559377*x^17/17! + 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! +...+ A276910(n)*x^n/n! +...
where
G( F(x) ) = x + 2*I*x^2/2! - 9*x^3/3! - 64*I*x^4/4! + 625*x^5/5! + 7776*I*x^6/6! - 117649*x^7/7! - 2097152*I*x^8/8! +...+ -n^(n-1)*(-i)^(n-1)*x^n/n! +...

Prog

(PARI) {a(n) = my(V=[1], A=x, G=x); for(i=1, n\2+1, V = concat(V, [0, 0]); G = sum(m=1, #V, V[m]*x^m/m!) +x*O(x^#V);
A = G*exp(I*G); V[#V] = -(#V)!/2 * polcoeff( subst( A, x, conj(A) ), #V) ); n!*(-I)^(n-1)*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A276910, A276912.

Sequence in context: A052809 A136631 A002435 * A104018 A100526 A200560

Adjacent sequences: A276908 A276909 A276910 * A276912 A276913 A276914

Keyword

sign

Author

Paul D. Hanna, Sep 22 2016

Extensions

Name replaced with simpler formula by Paul D. Hanna, Sep 06 2018

Status

approved