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A264226 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-8*x) ), with A(0) = 0.

%I A264226 #31 Nov 16 2015 18:28:58
%S A264226 1,4,26,184,1371,10524,82446,655624,5274581,42835444,350607226,
%T A264226 2888950904,23943016426,199450842504,1669044107916,14024053212624,
%U A264226 118272485941116,1000814156934384,8494876225031496,72307674880328544,617074982874821901,5278745007753158724,45256869801034564986,388802380782229815384,3346570416790776555756
%N A264226 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-8*x) ), with A(0) = 0.
%C A264226 Radius of convergence is r = 1/9, where r = r^2/(1-8*r), with A(r) = 1.
%C A264226 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 A264226 Paul D. Hanna, <a href="/A264226/b264226.txt">Table of n, a(n) for n = 1..300</a>
%F A264226 G.f. also satisfies:
%F A264226 (1) A(x) = -A( -x/(1-8*x) ).
%F A264226 (2) A( x/(1+4*x) ) = -A( -x/(1-4*x) ), an odd function.
%F A264226 (3) A( x/(1+4*x) )^2 = A( x^2/(1-16*x^2) ), an even function.
%F A264226 (4) A(x)^4 = A( x^4/((1-8*x)*(1-8*x-8*x^2)) ).
%F A264226 (5) [x^(2*n+1)] (x/A(x))^(2*n) = 0 for n>=0.
%F A264226 (6) [x^(2^n+k)] (x/A(x))^(2^n) = 0 for k=1..2^n-1, n>=1.
%F A264226 Given g.f. A(x), let F(x) denote the g.f. of A264414, then:
%F A264226 (7) A(x) = F(A(x))^2 * x/(1+16*x),
%F A264226 (8) A(x) = F(A(x)^2) * x/(1-4*x),
%F A264226 (9) A( x/(F(x)^2 - 16*x) ) = x,
%F A264226 (10) A( x/(F(x^2) + 4*x) ) = x,
%F A264226 where F(x)^2 = F(x^2) + 20*x.
%F A264226 Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = 0 for odd n.
%F A264226 Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.
%e A264226 G.f.: A(x) = x + 4*x^2 + 26*x^3 + 184*x^4 + 1371*x^5 + 10524*x^6 + 82446*x^7 + 655624*x^8 + 5274581*x^9 + 42835444*x^10 + 350607226*x^11 +...
%e A264226 where A(x)^2 = A(x^2/(1-8*x)).
%e A264226 RELATED SERIES.
%e A264226 A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 576*x^5 + 4890*x^6 + 41584*x^7 + 354232*x^8 + 3022592*x^9 + 25833819*x^10 + 221156920*x^11 + 1896267356*x^12 +...
%e A264226 (A(x)/x)^(1/2) = 1 + 2*x + 11*x^2 + 70*x^3 + 485*x^4 + 3522*x^5 + 26394*x^6 + 202332*x^7 + 1578140*x^8 + 12480040*x^9 + 99817421*x^10 + 805999682*x^11 +...
%e A264226 (A(x)/x)^(1/4) = 1 + x + 5*x^2 + 30*x^3 + 200*x^4 + 1411*x^5 + 10336*x^6 + 77775*x^7 + 597285*x^8 + 4661580*x^9 + 36864795*x^10 + 294769500*x^11 +...
%e A264226 A( x/(1+4*x) ) = x + 10*x^3 + 155*x^5 + 2750*x^7 + 52565*x^9 + 1055850*x^11 + 21979050*x^13 + 469891500*x^15 + 10252631420*x^17 + 227274091400*x^19 +...
%e A264226 A( x^2/(1-16*x^2) ) = x^2 + 20*x^4 + 410*x^6 + 8600*x^8 + 184155*x^10 + 4015500*x^12 + 88932750*x^14 + 1995785000*x^16 + 45286852565*x^18 +...
%e A264226 where A( x^2/(1-16*x^2) ) = A( x/(1+4*x) )^2.
%e A264226 Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
%e A264226 B(x) = 1 + 4*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 + 16953750*x^14 - 268652880*x^16 + 4365638550*x^18 +...+ A264414(n)*x^(2*n) +...
%e A264226 such that B(x) = F(x^2) + 4*x = F(x)^2 - 16*x and F(x) is the g.f. of A264414.
%o A264226 (PARI) {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-8*x +x*O(x^n))) ) ); polcoeff(A,n)}
%o A264226 for(n=1,30,print1(a(n),", "))
%Y A264226 Cf. A264414, A264224, A264225, A264227.
%K A264226 nonn
%O A264226 1,2
%A A264226 _Paul D. Hanna_, Nov 08 2015