OFFSET
1,2
COMMENTS
Radius of convergence is r = 1/9, where r = r^2/(1-8*r), with A(r) = 1.
Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. also satisfies:
(1) A(x) = -A( -x/(1-8*x) ).
(2) A( x/(1+4*x) ) = -A( -x/(1-4*x) ), an odd function.
(3) A( x/(1+4*x) )^2 = A( x^2/(1-16*x^2) ), an even function.
(4) A(x)^4 = A( x^4/((1-8*x)*(1-8*x-8*x^2)) ).
(5) [x^(2*n+1)] (x/A(x))^(2*n) = 0 for n>=0.
(6) [x^(2^n+k)] (x/A(x))^(2^n) = 0 for k=1..2^n-1, n>=1.
Given g.f. A(x), let F(x) denote the g.f. of A264414, then:
(7) A(x) = F(A(x))^2 * x/(1+16*x),
(8) A(x) = F(A(x)^2) * x/(1-4*x),
(9) A( x/(F(x)^2 - 16*x) ) = x,
(10) A( x/(F(x^2) + 4*x) ) = x,
where F(x)^2 = F(x^2) + 20*x.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.
EXAMPLE
G.f.: A(x) = x + 4*x^2 + 26*x^3 + 184*x^4 + 1371*x^5 + 10524*x^6 + 82446*x^7 + 655624*x^8 + 5274581*x^9 + 42835444*x^10 + 350607226*x^11 +...
where A(x)^2 = A(x^2/(1-8*x)).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 576*x^5 + 4890*x^6 + 41584*x^7 + 354232*x^8 + 3022592*x^9 + 25833819*x^10 + 221156920*x^11 + 1896267356*x^12 +...
(A(x)/x)^(1/2) = 1 + 2*x + 11*x^2 + 70*x^3 + 485*x^4 + 3522*x^5 + 26394*x^6 + 202332*x^7 + 1578140*x^8 + 12480040*x^9 + 99817421*x^10 + 805999682*x^11 +...
(A(x)/x)^(1/4) = 1 + x + 5*x^2 + 30*x^3 + 200*x^4 + 1411*x^5 + 10336*x^6 + 77775*x^7 + 597285*x^8 + 4661580*x^9 + 36864795*x^10 + 294769500*x^11 +...
A( x/(1+4*x) ) = x + 10*x^3 + 155*x^5 + 2750*x^7 + 52565*x^9 + 1055850*x^11 + 21979050*x^13 + 469891500*x^15 + 10252631420*x^17 + 227274091400*x^19 +...
A( x^2/(1-16*x^2) ) = x^2 + 20*x^4 + 410*x^6 + 8600*x^8 + 184155*x^10 + 4015500*x^12 + 88932750*x^14 + 1995785000*x^16 + 45286852565*x^18 +...
where A( x^2/(1-16*x^2) ) = A( x/(1+4*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 4*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 + 16953750*x^14 - 268652880*x^16 + 4365638550*x^18 +...+ A264414(n)*x^(2*n) +...
such that B(x) = F(x^2) + 4*x = F(x)^2 - 16*x and F(x) is the g.f. of A264414.
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = sqrt( subst(A, x, x^2/(1-8*x +x*O(x^n))) ) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 08 2015
STATUS
approved