A088674 Coefficients of the eigenfunction of a sequence transformation.

1, 3, 6, 45, 126, 750, 2796, 19389, 75894, 449562, 2027796, 12211794, 57895596, 332787324, 1677545304, 9766642077, 50378641830, 286825948194, 1529968671492, 8729259097158, 47374697101572, 269062276076868, 1484430536591592

Offset: 0

Author

Michael Somos, Oct 04 2003

Keywords

• nonn

Comments

G.f. A(x) satisfies A(x^2) = (A(x/2)-1)/x - A(x/2)^2/2.

B(x) := 1/(2*x) - x*A(x^2) satisfies B(x)^2 + 1 = B(2*x^2).

Define f(n, c) := x - Sum_{k>=0} a(k)/(2*x)^(2*k+1) where x = c^(2^n). Then A003095(n+1) = A004019(n) + 1 = f(n, 1.502836801...). Also, A062013(n) = f(n, 1.78050350...). - Michael Somos, Jun 07 2021

Examples

			G.f. = A(x) = 1 + 3*x + 6*x^2 + 45*x^3 + 126*x^4 + 750*x^5 + 2796*x^6 + ...
B(x) = 1/(2*x) - x - 3*x^3 - 6*x^5 - 45*x^7 - 126*x^9 - 750*x^11 - ... - _Michael Somos_, Jul 11 2019
		

Crossrefs

Cf. A003095, A004019, A062013.

Programs

Mathematica

a[ n_] := If[n < 0, 0, Module[{A = 1 + O[x], m = 2}, While[m < n + 2, m *= 2; A = (Normal[ 1/x - Sqrt[ 1/x^2 - 2/x - 2*(Normal[A] /. x -> x^2) + O[x]^(m - 2)]] /. x -> 2*x) + O[x]^(m - 1) //PowerExpand]; SeriesCoefficient[A, n]]]; (* Michael Somos, Jun 07 2021 *)

PARI

{a(n) = my(A, m); if( n < 0, 0, m=2; A = 1 + O(x); while( m < n+2, m*=2; A = subst(1/x - sqrt(2*(subst((1/2)/x - A, x, x^2) - 1/x)), x, 2*x)); polcoeff(A, n))};