A277404 E.g.f. A(x) satisfies: A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

1, 4, 36, 508, 10020, 253804, 7853076, 287078908, 12106864260, 578586544204, 30901130685876, 1823983173981148, 117911755067635620, 8284976875099852204, 628692318063511556436, 51240154266491883376828, 4464155216699369664399300, 414013560595951627772296204, 40722939746084736801890208756

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = 2*A143138(n) for n>1.

E.g.f. A(x) satisfies:

(1) A(x) = x + 2 * (exp( (A(x) + x)/2 ) - 1)^2.

(2) A(x) = -x + 2 * Series_Reversion( x - (exp(x)-1)^2 ).

(3) A(x) = x + 2 * Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n) / n!.

(4) A( log(1+x) - x^2 ) = log(1+x) + x^2.

Example

E.g.f.: A(x) = x + 4*x^2/2! + 36*x^3/3! + 508*x^4/4! + 10020*x^5/5! + 253804*x^6/6! + 7853076*x^7/7! + 287078908*x^8/8! + 12106864260*x^9/9! + 578586544204*x^10/10! +...
such that A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

Prog

(PARI) {a(n) = n!*polcoeff( -x + 2*serreverse( x - (exp(x +x*O(x^n)) - 1)^2 ), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A143138.

Sequence in context: A138435 A366337 A008546 * A024253 A052746 A145084

Adjacent sequences: A277401 A277402 A277403 * A277405 A277406 A277407

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2016

Status

approved