A275765 G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.

1, 2, 12, 106, 1148, 14156, 191400, 2775930, 42585412, 684496988, 11449962008, 198331811356, 3543990791480, 65136985937096, 1228531761076208, 23733123786608826, 468887742020767788, 9461919438245032500, 194817077269127033944, 4089069139317823277548, 87426000975842460304792, 1902787414323673070857528, 42133267254272433484761584, 948695717599714654940068604, 21712101305047777916075831096, 504865916349551192319293625016

Offset: 1

Author

Paul D. Hanna, Aug 31 2016

Keywords

• nonn

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +...
such that A(x - A(x)^2) = x + A(x)^2.
RELATED SERIES.
Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +...
which equals (A(x) + x)/2.
A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +...
which equals sqrt( (A(x) - x)/2 ).
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +...
then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A277295, A213591, A276364, A276360, A276361, A276362, A276363.

Programs

Mathematica

m = 26; A[_] = 0;
Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 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-F^2) - F^2,#A) );A[n]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:

(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^2.

(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^2).

(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.

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

a(n) = Sum_{k=0..n-1} A277295(n,k)*2^(n-k).