A277403 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.

1, 2, 10, 90, 1190, 20930, 462070, 12326790, 386855630, 14000898310, 575440398330, 26532920708070, 1358954912773010, 76682330257445570, 4734315243483414890, 317932511564758225170, 23106045191162625194230, 1809303767549542227341490, 152057767850058496005946030, 13668688227104664304597942910, 1310201986290043690952261887230, 133552478071366935949713096470670, 14440878313638992240490923468851610

Offset

1,2

Comments

a(n) is divisible by 10 for n>2 (conjecture).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

Let G(x) = Integral A(x) dx, then e.g.f. A(x) also satisfies:

(1) A( (A(x) + x)/2 ) = (A'(x) - 1)/(A'(x) + 1).

(2) A(x) = x + 2 * G( (A(x) + x)/2 ).

(3) A(x) = -x + 2 * Series_Reversion(x - G(x)).

(4) R(x) = -x + 2 * Series_Reversion(x + G(x)), where R(A(x)) = x.

(5) R( sqrt( x/2 - R(x)/2 ) ) = x/2 + R(x)/2, where R(A(x)) = x.

a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^(n-k-1).

Example

E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 90*x^4/4! + 1190*x^5/5! + 20930*x^6/6! + 462070*x^7/7! + 12326790*x^8/8! + 386855630*x^9/9! + 14000898310*x^10/10! +...
such that
A(x - Integral A(x) dx) = x + x^2/2! + 2*x^3/3! + 10*x^4/4! + 90*x^5/5! + 1190*x^6/6! + 20930*x^7/7! + 462070*x^8/8! +...+ a(n)*x^(n+1)/(n+1)! +...
which equals x + Integral A(x) dx.
RELATED SERIES.
Let G(x) = Integral A(x) dx, then
G( (A(x) + x)/2 ) = x^2/2! + 5*x^3/3! + 45*x^4/4! + 595*x^5/5! + 10465*x^6/6! + 231035*x^7/7! + 6163395*x^8/8! +...+ a(n)/2*x^n/n! +...
so that A(x) = x + 2 * G( (A(x) + x)/2 ).
A( (A(x) + x)/2 ) = x + 3*x^2/2! + 21*x^3/3! + 241*x^4/4! + 3885*x^5/5! + 81185*x^6/6! + 2093735*x^7/7! + 64463245*x^8/8! + 2313446975*x^9/9! + 95044136915*x^10/10! +...
which equals (A'(x) - 1)/(A'(x) + 1).

Mathematica

m = 24; A[_] = 0;
Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 2 G[(A[x] + x)/2] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m-1]! // Rest (* Jean-François Alcover, Oct 20 2019 *)

Prog

(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F = x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - intformal(F)) - intformal(F), #A) ); n!*A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A277410, A210949, A279843.

Sequence in context: A366268 A198434 A326089 * A179423 A320962 A067550

Adjacent sequences: A277400 A277401 A277402 * A277404 A277405 A277406

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2016

Status

approved