A264225 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-6*x) ), with A(0) = 0.

1, 3, 15, 81, 462, 2718, 16344, 99900, 618567, 3870909, 24441021, 155510523, 996109245, 6418243575, 41572149615, 270536350545, 1767990955980, 11598120859860, 76347126498420, 504148079084940, 3338585176489560, 22166530404950520, 147525638070221640, 983978335278966456, 6576191509703182677, 44031626057441376423

Offset: 1

Author

Paul D. Hanna, Nov 08 2015

Keywords

• nonn

Comments

Radius of convergence is r = 1/7, where r = r^2/(1-6*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).

Examples

			G.f.: A(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 + 24441021*x^11 + 155510523*x^12 +...
where A(x)^2 = A(x^2/(1-6*x)).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 252*x^5 + 1635*x^6 + 10638*x^7 + 69417*x^8 + 454248*x^9 + 2980614*x^10 + 19609380*x^11 + 129337686*x^12 +...
A( x/(1+3*x) ) = x + 6*x^3 + 57*x^5 + 630*x^7 + 7584*x^9 + 96552*x^11 + 1277937*x^13 + 17393454*x^15 + 241666275*x^17 + 3410638362*x^19 + 48723929721*x^21 +...
A( x^2/(1-9*x^2) ) = x^2 + 12*x^4 + 150*x^6 + 1944*x^8 + 25977*x^10 + 355932*x^12 + 4975974*x^14 + 70684920*x^16 + 1016911392*x^18 + 14778827136*x^20 +...
where A( x^2/(1-9*x^2) ) = A( x/(1+3*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 3*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 +...+ A264413(n)*x^(2*n) +...
such that B(x) = F(x^2) + 3*x = F(x)^2 - 9*x and F(x) is the g.f. of A264413.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A264413, A264233, A264224, A264226, A264227.

Programs

Mathematica

max = 25; For[A = x; i = 1, i <= max, i++, A = Sqrt[Normal[A] /. x -> x^2/(1 - 6*x + x*O[x]^max)]]; CoefficientList[A, x] // Rest (* Jean-François Alcover, Nov 22 2016 *)

PARI

{a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-6*x +x*O(x^n))) ) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. also satisfies:

(1) A(x) = -A( -x/(1-6*x) ).

(2) A( x/(1+3*x) ) = -A( -x/(1-3*x) ), an odd function.

(3) A( x/(1+3*x) )^2 = A( x^2/(1-9*x^2) ), an even function.

(4) A(x)^4 = A( x^4/((1-6*x)*(1-6*x-6*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 A264413, then:

(7) A(x) = F(A(x))^2 * x/(1+9*x),

(8) A(x) = F(A(x)^2) * x/(1-3*x),

(9) A( x/(F(x)^2 - 9*x) ) = x,

(10) A( x/(F(x^2) + 3*x) ) = x,

where F(x)^2 = F(x^2) + 12*x.

Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * a(k+1) = 0 for odd n.

Sum_{k=0..n} binomial(n,k) * (-6)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.