A277403 - OEIS

%I #21 Oct 20 2019 09:38:03

%S 1,2,10,90,1190,20930,462070,12326790,386855630,14000898310,

%T 575440398330,26532920708070,1358954912773010,76682330257445570,

%U 4734315243483414890,317932511564758225170,23106045191162625194230,1809303767549542227341490,152057767850058496005946030,13668688227104664304597942910,1310201986290043690952261887230,133552478071366935949713096470670,14440878313638992240490923468851610

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

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

%H Paul D. Hanna, <a href="/A277403/b277403.txt">Table of n, a(n) for n = 1..200</a>

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

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

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

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

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

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

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

%e 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! +...

%e such that

%e 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)! +...

%e which equals x + Integral A(x) dx.

%e RELATED SERIES.

%e Let G(x) = Integral A(x) dx, then

%e 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! +...

%e so that A(x) = x + 2 * G( (A(x) + x)/2 ).

%e 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! +...

%e which equals (A'(x) - 1)/(A'(x) + 1).

%t m = 24; A[_] = 0;

%t Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 2 G[(A[x] + x)/2] + O[x]^m // Normal, {m}];

%t CoefficientList[A[x], x]*Range[0, m-1]! // Rest (* _Jean-François Alcover_, Oct 20 2019 *)

%o (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]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A277410, A210949, A279843.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 14 2016