|
| |
|
|
A260650
|
|
G.f. satisfies: A(x)^2 = A( x^2/(1-4*x)^2 ).
|
|
3
|
|
|
|
1, 4, 18, 88, 455, 2444, 13486, 75912, 433935, 2511388, 14684422, 86611848, 514704064, 3078845696, 18523994024, 112026315616, 680626958899, 4152411174284, 25428402204982, 156247439709832, 963048223399984, 5952595420121536, 36887847899094888, 229132114803540320, 1426367728966653535, 8897049258366111004
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Radius of convergence is r = (9 - sqrt(17))/32 where r = r^2/(1-4*r)^2 with A(r) = 1.
Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers, A000108.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. satisfies:
(1) A(x) = -A( -x/(1-8*x) ).
(2) A(x^2) = A( x/(1+4*x) )^2 = A( -x/(1-4*x) )^2.
(3) A( x/(1+2*x)^2 ) = -A( -x/(1-2*x)^2 ), an odd function.
(4) A( x/(1+2*x)^2 )^2 = A( x^2/(1+4*x^2)^2 ), an even function.
|
|
|
EXAMPLE
|
G.f.: A(x) = x + 4*x^2 + 18*x^3 + 88*x^4 + 455*x^5 + 2444*x^6 + 13486*x^7 + 75912*x^8 + 433935*x^9 + 2511388*x^10 + 14684422*x^11 + 86611848*x^12 +...
where A( x^2/(1-4*x)^2 ) = A(x)^2,
A( x^2/(1-4*x)^2 ) = x^2 + 8*x^3 + 52*x^4 + 320*x^5 + 1938*x^6 + 11696*x^7 + 70648*x^8 + 427776*x^9 + 2597831*x^10 + 15824664*x^11 + 96687516*x^12 +...
Also, A( x/(1+4*x) ) = A(x^2)^(1/2),
A( x/(1+4*x) ) = x + 2*x^3 + 7*x^5 + 30*x^7 + 143*x^9 + 726*x^11 + 3840*x^13 + 20904*x^15 + 116275*x^17 + 657798*x^19 + 3772912*x^21 + 21890152*x^23 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 - 233*x^16 + 668*x^18 - 1949*x^20 + 5802*x^22 +...+ A107087(n)*x^(2*n) +...
such that B(x) = F(x^2) + 4*x = F(x)^2 where F(x) is the g.f. of A107087.
|
|
|
PROG
|
(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x +x*O(x^n))^2) ) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
