A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2*x) * exp( Integral A(x) dx ), where the constant of integration is zero.

1, 1, 4, 20, 126, 972, 8876, 93580, 1119328, 14986944, 222184136, 3614288272, 64022264176, 1226914925840, 25295189791296, 558317369479616, 13136590271813856, 328243850207690432, 8680766764223956416, 242245419192494844096, 7113910552105144027136, 219304957649505551899136, 7081169542830272102170752, 238996807468258679150596352

Offset: 0

Author

Paul D. Hanna, Jan 27 2016

Keywords

• nonn

Comments

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 20*x^3/3! + 126*x^4/4! + 972*x^5/5! + 8876*x^6/6! + 93580*x^7/7! + 1119328*x^8/8! + 14986944*x^9/9! + 222184136*x^10/10! +...
such that log(A(x)) = Integral B(x) dx
where B(x) = 1 + 3*x + 10*x^2/2! + 40*x^3/3! + 206*x^4/4! + 1384*x^5/5! + 11644*x^6/6! + 116868*x^7/7! + 1353064*x^8/8! + 17693072*x^9/9! + 257570280*x^10/10! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) - 2,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) + 2*log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx + 2*x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + 2*x) dx:
Integral 1/(exp(x) + 2*x) dx  =  x - 3*x^2/2! + 17*x^3/3! - 145*x^4/4! + 1649*x^5/5! - 23441*x^6/6! + 399865*x^7/7! - 7957881*x^8/8! + 180997857*x^9/9! - 4631289697*x^10/10! +...
so that A( Integral 1/(exp(x) + 2*x) dx ) = exp(x).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A266328, A266329.

Programs

Mathematica

a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ 2^(j - 1) a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *)

PARI

{a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( 2 + A ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + 2*x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

(1) A(x) = exp( Integral A(x) + 2*log(A(x)) dx ).

(2) A(x) = A'(x)/A(x) - 2*log(A(x)).

(3) log(A(x)) = exp(2*x) * Integral exp(-2*x)*A(x) dx.

(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + 2*x) dx ) ).

a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(2*x + exp(x)) dx = 1.4650202775490107369040248583790383461628786237838809798971... - Vaclav Kotesovec, Aug 21 2017