A357547 - OEIS

%I #18 Dec 04 2022 07:34:08

%S 1,2,9,38,176,832,4039,19938,99861,506042,2590099,13370898,69540016,

%T 364028992,1916585714,10142059868,53911982971,287736310102,

%U 1541243386819,8282387269058,44638363790176,241216694913632,1306608966475854,7092980525443588,38581011402034156

%N a(n) = coefficient of x^n in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

%C Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = 1.

%C Related identities:

%C (1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).

%C (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).

%C More generally, if

%C F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),

%C then

%C F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );

%C here, a = 2, b = 6.

%H Paul D. Hanna, <a href="/A357547/b357547.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

%F (1) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ).

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

%F (3.a) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

%F (3.b) A(x)^2 = -A( -x^2/(1 - 4*x - 8*x^2) ).

%F (4.a) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ).

%F (4.b) A( x/(1 + 2*x) )^2 = -A( -x^2/(1 - 12*x^2) ).

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

%e G.f.: A(x) = x + 2*x^2 + 9*x^3 + 38*x^4 + 176*x^5 + 832*x^6 + 4039*x^7 + 19938*x^8 + 99861*x^9 + 506042*x^10 + 2590099*x^11 + 13370898*x^12 + ...

%e where A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 112*x^5 + 585*x^6 + 3052*x^7 + 16018*x^8 + 84384*x^9 + 446384*x^10 + 2370240*x^11 + 12631104*x^12 + ...

%e (x*A(x))^(1/2) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + ... + A357785(n)*x^n + ...

%e x/Series_Reversion(A(x)) = 1 + 2*x + 5*x^2 - 10*x^4 + 50*x^6 - 305*x^8 + 2025*x^10 - 14400*x^12 + 107500*x^14 - 829415*x^16 + 6559700*x^18 - 52908950*x^20 + ...

%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),

%o A = sqrt( subst(A, x, x^2/(1 - 4*x - 4*x^2 +x*O(x^n)) ) )

%o ); polcoeff(A, n)}

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

%Y Cf. A357785, A264224, A274483, A274484, A357548.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 01 2022