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

Author

Paul D. Hanna, Oct 14 2016

Keywords

• nonn

Comments

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A277410, A210949, A279843.

Programs

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 *)

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), ", "))

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).