A274478 G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 2*x^2) ).

1, 1, 3, 7, 20, 56, 166, 498, 1530, 4762, 15022, 47862, 153859, 498239, 1623779, 5321059, 17520994, 57937106, 192304222, 640446358, 2139414409, 7166431909, 24065926653, 81003492725, 273229977460, 923438683996, 3126674842896, 10604713671208, 36025426127382, 122566140787390, 417584644921806, 1424610537707166, 4866239784751346, 16642071212737394, 56978489024931038, 195289731964727862, 670023314236521396, 2301024202252503308, 7909580344156028160

Offset: 1

Author

Paul D. Hanna, Jul 26 2016

Keywords

• nonn

Comments

Radius of convergence of g.f. A(x) is r = (sqrt(17) - 3)/4 where r = r^2/(1-2*r-2*r^2) with A(r) = 1.

Compare g.f. with the identities:

(1) F(x)^2 = F( x^2/(1 - 2*x + 2*x^2) ) when F(x) = x/(1-x).

(2) M(x)^2 = M( x^2/(1 - 2*x) ) when M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 56*x^6 + 166*x^7 + 498*x^8 + 1530*x^9 + 4762*x^10 + 15022*x^11 + 47862*x^12 +...
such that A( x^2/(1-2*x-2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 20*x^5 + 63*x^6 + 194*x^7 + 613*x^8 + 1944*x^9 + 6236*x^10 + 20136*x^11 + 65496*x^12 + 214272*x^13 + 704774*x^14 + 2328852*x^15 +...
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - x^2 - x^3 + 3*x^4 - 7*x^6 + 4*x^7 + 15*x^8 - 16*x^9 - 32*x^10 + 51*x^11 + 69*x^12 - 153*x^13 - 148*x^14 + 445*x^15 + 315*x^16 +...
which is related to A107087 by:
x/Series_Reversion(A(x)) = 1 + 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 +...+ A107087(n)*x^(2*n) +...
The g.f. G(x) of A107087 begins:
G(x) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 + 668*x^9 - 1949*x^10 + 5802*x^11 - 17503*x^12 +...
where G(x)^2 = G(x^2) + 4*x.
Also, we have A(x/(1 + x + 2*x^2))^2 = A(x^2/(1 + x^2 + 4*x^4)), where the series begin:
A(x/(1 + x + 2*x^2)) = x - x^5 - x^9 + 8*x^13 - 13*x^17 - 8*x^21 - x^25 + 307*x^29 + 135*x^33 - 9641*x^37 + 36869*x^41 +...
A(x^2/(1 + x^2 + 4*x^4)) = x^2 - 2*x^6 - x^10 + 18*x^14 - 41*x^18 - 6*x^22 + 104*x^26 + 424*x^30 - 301*x^34 - 19974*x^38 + 97752*x^42 +...
which is equal to A(x/(1 + x + 2*x^2))^2.
		

Links

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

Crossrefs

Cf. A107087, A274479, A274484, A260650.

Programs

PARI

{a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-2*x-2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: A( x/(1 + x + 2*x^2) )^2 = A( x^2/(1 + x^2 + 4*x^4) ).

Let G(x) denote the g.f. of A107087, where G(x)^2 = G(x^2) + 4*x, then g.f. A(x) satisfies:

(1) A(x) = x/(1-x) * G( A(x)^2 ),

(2) G(x^2) = x/Series_Reversion(A(x)) - x,

(3) A( x/(G(x^2) + x) ) = x,

(4) A(x)^2/(G(A(x)^4) + A(x)^2) = x^2/(1 - 2*x - 2*x^2).