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%I #43 Nov 30 2015 14:05:19
%S 1,2,7,26,103,422,1774,7604,33109,146042,651256,2931392,13301038,
%T 60775340,279393742,1291311620,5996491666,27962898020,130883946751,
%U 614664907706,2895279687655,13674609742598,64744203198388,307221794213768,1460778188820220,6958635514922552,33205258829750809,158699556581760134
%N G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-4*x) ), with A(0) = 0.
%C Radius of convergence is r = 1/5, where r = r^2/(1-4*r), with A(r) = 1.
%C Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).
%H Paul D. Hanna, <a href="/A264224/b264224.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. also satisfies:
%F (1) A(x) = -A( -x/(1-4*x) ).
%F (2) A( x/(1+2*x) ) = -A( -x/(1-2*x) ), an odd function.
%F (3) A( x/(1+2*x) )^2 = A( x^2/(1-4*x^2) ), an even function.
%F (4) A(x)^4 = A( x^4/((1-4*x)*(1-4*x-4*x^2)) ).
%F (5) [x^(2*n+1)] (x/A(x))^(2*n) = 0 for n>=0.
%F (6) [x^(2^n+k)] (x/A(x))^(2^n) = 0 for k=1..2^n-1, n>=1.
%F Given g.f. A(x), let F(x) denote the g.f. of A264412, then:
%F (7) A(x) = F(A(x))^2 * x/(1+4*x),
%F (8) A(x) = F(A(x)^2) * x/(1-2*x),
%F (9) A( x/(F(x)^2 - 4*x) ) = x,
%F (10) A( x/(F(x^2) + 2*x) ) = x,
%F where F(x)^2 = F(x^2) + 6*x.
%F Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * a(k+1) = 0 for odd n.
%F Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.
%F Sum_{k=0..n} binomial(n,k) * (+4)^(n-k) * a(k+1) = A264232(n+1) for n>=0.
%F Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264232(n+1) for n>=0.
%e G.f.: A(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 103*x^5 + 422*x^6 + 1774*x^7 + 7604*x^8 + 33109*x^9 + 146042*x^10 + 651256*x^11 + 2931392*x^12 +...
%e where A(x)^2 = A(x^2/(1-4*x)).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 4*x^3 + 18*x^4 + 80*x^5 + 359*x^6 + 1620*x^7 + 7354*x^8 + 33568*x^9 + 154023*x^10 + 710156*x^11 + 3289142*x^12 + 15297744*x^13 +...
%e sqrt(A(x)/x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 144*x^5 + 582*x^6 + 2418*x^7 + 10266*x^8 + 44353*x^9 + 194395*x^10 +...+ A264231(n)*x^n +...
%e A( x/(1+2*x) ) = x + 3*x^3 + 15*x^5 + 90*x^7 + 597*x^9 + 4212*x^11 + 30942*x^13 + 233766*x^15 + 1802706*x^17 + 14122359*x^19 + 112033791*x^21 + 898024320*x^23 +...
%e A( x^2/(1-4*x^2) ) = x^2 + 6*x^4 + 39*x^6 + 270*x^8 + 1959*x^10 + 14706*x^12 + 113166*x^14 + 887004*x^16 + 7050837*x^18 + 56672622*x^20 + 459646488*x^22 +...
%e where A( x^2/(1-4*x^2) ) = A( x/(1+2*x) )^2.
%e Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
%e B(x) = 1 + 2*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 +...+ A264412(n)*x^(2*n) +...
%e such that B(x) = F(x^2) + 2*x = F(x)^2 - 4*x and F(x) is the g.f. of A264412.
%e PARTICULAR VALUES.
%e A(1/5) = 1.
%e A(-1/5) = -A(1/9) = -0.15262256991492310976978497600904...
%e A(1/6)^2 = A(1/12) = 0.10315964246752710052686298695713...
%e A(1/6)^4 = A(1/96) = 0.01064191183402802084987998396215...
%e A(1/7)^2 = A(1/21) = 0.053075120978549663441827849989065...
%e A(1/7)^4 = A(1/357) = 0.002816968466887682583828696137137...
%e A(1/8)^2 = A(1/32) = 0.033445065874191867268119916059631...
%e A(1/8)^4 = A(1/896) = 0.001118572431329033410718706838540...
%e A(1/9)^2 = A(1/45) = 0.0232936488474355927381514600230212...
%o (PARI) {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-4*x +x*O(x^n))) ) ); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A264231, A264232, A264412, A264225, A264226, A264227.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Nov 08 2015
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