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

0,2

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

Links

Table of n, a(n) for n=0..22.

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A003095, A004019, A062013.

Sequence in context: A092198 A133005 A009581 * A203434 A076170 A137775

Adjacent sequences: A088671 A088672 A088673 * A088675 A088676 A088677

Keyword

nonn

Author

Michael Somos, Oct 04 2003

Status

approved