|
| |
|
|
A357548
|
|
a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
|
|
4
|
|
|
|
1, 2, 11, 50, 261, 1362, 7344, 40112, 222338, 1245476, 7043605, 40153390, 230518723, 1331576430, 7733934030, 45138530004, 264596552838, 1557101158092, 9195520745412, 54477134410680, 323668083179382, 1928047124332764, 11512382184408072, 68889282756213840
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = 1.
Related identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );
here, a = 2, b = 8.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ).
(5) A( x/(1 + 4*x) )^2 = A( x^2/(1 + 4*x - 8*x^2) ).
|
|
|
EXAMPLE
|
G.f.: A(x) = x + 2*x^2 + 11*x^3 + 50*x^4 + 261*x^5 + 1362*x^6 + 7344*x^7 + 40112*x^8 + 222338*x^9 + 1245476*x^10 + 7043605*x^11 + 40153390*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 144*x^5 + 843*x^6 + 4868*x^7 + 28378*x^8 + 165664*x^9 + 971013*x^10 + 5708132*x^11 + 33660362*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + ... + A357786(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 7*x^2 - 21*x^4 + 147*x^6 - 1260*x^8 + 11907*x^10 - 120540*x^12 + 1279047*x^14 - 14029428*x^16 + 157788183*x^18 + ...
|
|
|
PROG
|
(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
