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

0,3

Comments

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

Links

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

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

Example

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).

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A266328, A266329.

Sequence in context: A349603 A285868 A361548 * A135886 A007550 A361558

Adjacent sequences: A266487 A266488 A266489 * A266491 A266492 A266493

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2016

Status

approved