|
| |
|
|
A368633
|
|
Expansion of g.f. A(x) satisfying A(x) = 1 + 2*x*A(x)^2 - x*A(-x)^2.
|
|
6
|
|
|
|
1, 1, 6, 13, 114, 290, 2892, 7901, 84090, 239222, 2648244, 7732914, 87894324, 261371940, 3027588120, 9125058525, 107215635402, 326501869166, 3879094785060, 11910103389734, 142766337272988, 441265565242268, 5328172865489448, 16559430499708018, 201171901999797924
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + 2*x*A(x)^2 - x*A(-x)^2.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 3*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + 3*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - 3*x*(A(x) - A(-x))).
(3.a) A(x) = (1 - sqrt(1-8*x + 8*x^2*A(-x)^2)) / (4*x).
(3.b) A(-x) = (sqrt(1+8*x + 8*x^2*A(x)^2) - 1) / (4*x).
(4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(5) 0 = (1-x) - (1-4*x)*A(x) - 2*x*(1+3*x)*A(x)^2 + 12*x^2*A(x)^3 - 9*x^3*A(x)^4.
(6) x = (1 + 4*x*A(x) - 6*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/2.
(7) A(x) = (1/x)*Series_Reversion( (1 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2))/2 ).
(8.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
(8.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.
Conjecture D-finite with recurrence +375*n*(127034*n -380695)*(n-1)*(n+1) *a(n) -50*n*(n-1) *(1761680*n^2 -13025595*n +22423399)*a(n-1) -24*(n-1) *(85874984*n^3 -429099788*n^2 +603573743*n -272338575)*a(n-2) +8*(476358272*n^4 -5427553976*n^3 +22342502584*n^2 -39872302249*n +26255347914)*a(n-3) +864*(2*n-7) *(508136*n^3 -2793120*n^2 +5307886*n -3864543)*a(n-4) -1152*(n-4) *(352336*n-843439) *(2*n-7) *(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 24 2024
|
|
|
EXAMPLE
|
G.f. A(x) = 1 + x + 6*x^2 + 13*x^3 + 114*x^4 + 290*x^5 + 2892*x^6 + 7901*x^7 + 84090*x^8 + 239222*x^9 + 2648244*x^10 + 7732914*x^11 + 87894324*x^12 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 13*x^2 + 38*x^3 + 290*x^4 + 964*x^5 + 7901*x^6 + 28030*x^7 + 239222*x^8 + 882748*x^9 + 7732914*x^10 + 29298108*x^11 + 261371940*x^12 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 13*x^3 + 290*x^5 + 7901*x^7 + 239222*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 13*x^2 + 290*x^4 + 7901*x^6 + 239222*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 6*x^2 + 114*x^4 + 2892*x^6 + 84090*x^8 + 2648244*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 38*x^3 + 964*x^5 + 28030*x^7 + 882748*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 3*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071187457698....
|
|
|
PROG
|
(PARI) {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + 3*x*(A^2 - B^2)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A); A = (1/x)*serreverse( (1 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/2 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
