A274484 G.f. satisfies: A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).

1, 2, 6, 20, 71, 262, 994, 3852, 15183, 60686, 245410, 1002300, 4128448, 17129920, 71529800, 300355184, 1267386163, 5371101382, 22850230642, 97546995260, 417717017392, 1793765580704, 7722405668232, 33323153856880, 144099312039391, 624347587536782, 2710036186345914, 11782865084403212, 51310167663855675, 223762749750806942, 977155903597684074, 4272633455348970588, 18704696346822470087, 81978422471165944654

Offset

1,2

Comments

Radius of convergence of g.f. A(x) is r = (5 - sqrt(17))/4 where r = r^2/(1-4*r+2*r^2) with A(r) = 1.

Compare g.f. with the 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) ).

Links

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

Formula

G.f. A(x) satisfies:

(1) A(x) = -A( -x/(1 - 4*x) ). - Paul D. Hanna, Nov 30 2022

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

(3) A( x/(1 + 2*x + 3*x^2) )^2 = A( x^2/(1 + 4*x^2 + 9*x^4) ).

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

(5) A( x/(1 - 2*x) )^2 = A( x^2/(1 - 8*x + 14*x^2) ).

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

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

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

(8) G(x) = sqrt( x/Series_Reversion(A(x)) + 2*x ),

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

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

(11) A( x/(G(x^2) + 2*x) ) = x,

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

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 71*x^5 + 262*x^6 + 994*x^7 + 3852*x^8 + 15183*x^9 + 60686*x^10 + 245410*x^11 + 1002300*x^12 +...
such that A( x^2/(1-4*x+2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 258*x^6 + 1048*x^7 + 4288*x^8 + 17664*x^9 + 73223*x^10 + 305292*x^11 + 1279632*x^12  + 5389632*x^13 + 22800926*x^14 +...
The g.f. of A260650, F(x), begins:
A( x/(1 - 2*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 +...
and satisfies: F(x)^2 = F( x^2/(1 - 4*x)^2 ).
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - 2*x^2 + 2*x^3 - 3*x^5 + 4*x^6 - 2*x^7 + 2*x^9 - 10*x^10 + 18*x^11 - 39*x^13 + 28*x^14 + 40*x^15 - 142*x^17 - 84*x^18 + 620*x^19 - 1735*x^21 + 260*x^22 + 4532*x^23 +...
which is related to A107087 by:
x/Series_Reversion(A(x)) = 1 + 2*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 + 2*x + 3*x^2))^2 = A(x^2/(1 + 4*x^2 + 9*x^4)), where the series begin:
A(x/(1 + 2*x + 3*x^2)) = x - x^3 - 2*x^5 + 6*x^7 - x^9 - 3*x^11 - 30*x^13 - 66*x^15 + 715*x^17 - 747*x^19 - 4028*x^21 + 9424*x^23 + 8790*x^25 +...
A(x^2/(1 + 4*x^2 + 9*x^4)) = x^2 - 2*x^4 - 3*x^6 + 16*x^8 - 10*x^10 - 28*x^12 - 14*x^14 - 72*x^16 + 1647*x^18 - 3014*x^20 - 10145*x^22 + 38784*x^24 +...
which is equal to A(x/(1 + 2*x + 3*x^2))^2.

Prog

(PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x+2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A107087, A260650, A264224, A274483, A274478, A274479.

Cf. A264225, A357547, A357548.

Sequence in context: A000707 A129777 A108600 * A128729 A006027 A049124

Adjacent sequences: A274481 A274482 A274483 * A274485 A274486 A274487

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2016

Status

approved