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%I #23 Dec 01 2022 18:55:58
%S 1,2,6,20,71,262,994,3852,15183,60686,245410,1002300,4128448,17129920,
%T 71529800,300355184,1267386163,5371101382,22850230642,97546995260,
%U 417717017392,1793765580704,7722405668232,33323153856880,144099312039391,624347587536782,2710036186345914,11782865084403212,51310167663855675,223762749750806942,977155903597684074,4272633455348970588,18704696346822470087,81978422471165944654
%N G.f. satisfies: A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).
%C 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.
%C Compare g.f. with the 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) ).
%H Paul D. Hanna, <a href="/A274484/b274484.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = -A( -x/(1 - 4*x) ). - _Paul D. Hanna_, Nov 30 2022
%F (2) A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).
%F (3) A( x/(1 + 2*x + 3*x^2) )^2 = A( x^2/(1 + 4*x^2 + 9*x^4) ).
%F (4) A( x/(1 + 2*x) )^2 = x * A( x/(1 - 2*x) ).
%F (5) A( x/(1 - 2*x) )^2 = A( x^2/(1 - 8*x + 14*x^2) ).
%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 (6) A(x) = x/(1-2*x) * G( A(x)^2 ),
%F (7) A(x) = Series_Reversion( x/(G(x)^2 - 2*x) ),
%F (8) G(x) = sqrt( x/Series_Reversion(A(x)) + 2*x ),
%F (9) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
%F (10) A( x/(G(x)^2 - 2*x) ) = x,
%F (11) A( x/(G(x^2) + 2*x) ) = x,
%F (12) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x + 2*x^2).
%e 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 +...
%e such that A( x^2/(1-4*x+2*x^2) ) = A(x)^2.
%e RELATED SERIES.
%e 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 +...
%e The g.f. of A260650, F(x), begins:
%e 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 +...
%e and satisfies: F(x)^2 = F( x^2/(1 - 4*x)^2 ).
%e The series reversion of the g.f. A(x) begins:
%e 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 +...
%e which is related to A107087 by:
%e 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) +...
%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 + 2*x + 3*x^2))^2 = A(x^2/(1 + 4*x^2 + 9*x^4)), where the series begin:
%e 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 +...
%e 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 +...
%e which is equal to A(x/(1 + 2*x + 3*x^2))^2.
%o (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)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A107087, A260650, A264224, A274483, A274478, A274479.
%Y Cf. A264225, A357547, A357548.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 27 2016
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