A274478 - OEIS

%I #16 Aug 01 2016 17:59:00

%S 1,1,3,7,20,56,166,498,1530,4762,15022,47862,153859,498239,1623779,

%T 5321059,17520994,57937106,192304222,640446358,2139414409,7166431909,

%U 24065926653,81003492725,273229977460,923438683996,3126674842896,10604713671208,36025426127382,122566140787390,417584644921806,1424610537707166,4866239784751346,16642071212737394,56978489024931038,195289731964727862,670023314236521396,2301024202252503308,7909580344156028160

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

%C 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.

%C Compare g.f. with the identities:

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

%C (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).

%H Paul D. Hanna, <a href="/A274478/b274478.txt">Table of n, a(n) for n = 1..300</a>

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

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

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

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

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

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

%e 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 +...

%e such that A( x^2/(1-2*x-2*x^2) ) = A(x)^2.

%e RELATED SERIES.

%e 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 +...

%e The series reversion of the g.f. A(x) begins:

%e 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 +...

%e which is related to A107087 by:

%e 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) +...

%e The g.f. G(x) of A107087 begins:

%e 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 +...

%e where G(x)^2 = G(x^2) + 4*x.

%e Also, we have A(x/(1 + x + 2*x^2))^2 = A(x^2/(1 + x^2 + 4*x^4)), where the series begin:

%e 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 +...

%e 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 +...

%e which is equal to A(x/(1 + x + 2*x^2))^2.

%o (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)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A107087, A274479, A274484, A260650.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 26 2016