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A369082 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.

%I A369082 #15 Jan 15 2024 01:48:12
%S A369082 1,1,1,3,4,15,22,91,140,612,969,4389,7084,32890,53820,254475,420732,
%T A369082 2017356,3362260,16301164,27343888,133767543,225568798,1111731933,
%U A369082 1882933364,9338434700,15875338990,79155435870,134993766600,676196049060,1156393243320,5815796869995
%N A369082 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.
%C A369082 Equals the interleaving of sequences A002293 and A006632.
%H A369082 Paul D. Hanna, <a href="/A369082/b369082.txt">Table of n, a(n) for n = 0..1000</a>
%F A369082 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A369082 (1.a) A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.
%F A369082 (1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
%F A369082 (2.a) (A(x) + A(-x))/2 = 1 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
%F A369082 (2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
%F A369082 (2.c) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))/2).
%F A369082 (2.d) (A(x) + A(-x))/2 = F(x^2) where F(x) = 1 + x*F(x)^4 (cf. A002293).
%F A369082 (2.e) (A(x) - A(-x))/2 = x*F(x^2)^3 where F(x) = 1 + x*F(x)^4 (cf. A006632).
%F A369082 (3.a) A(x) = (1 - sqrt(1 - 4*x + 2*x*A(-x) + x^2*A(-x)^2))/x.
%F A369082 (3.b) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2))/(2*x).
%F A369082 (4.a) 0 = (1+4*x) - (1+6*x)*A(x) + (3+4*x)*x*A(x)^2 - 3*x^2*A(x)^3 + x^3*A(x)^4.
%F A369082 (4.b) x = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - (1 - 6*x*A(x) + 4*x^2*A(x)^2))/8.
%F A369082 (5.a) A(x) = (1/x)*Series_Reversion( (sqrt(1 + 4*x - 4*x^2) - (1 - 6*x + 4*x^2))/8 ).
%F A369082 (5.b) (A(x) + A(-x))/2 = (1/x)*Series_Reversion( x/C(x^2) ) =  where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan sequence (A000108).
%F A369082 (5.c) (A(x) - A(-x))/2 = Series_Reversion( x*D(-x^2)^3 ) where D(x) = 1 + x*D(x)^3 (cf. A001764).
%F A369082 (6.a) a(2*n) = binomial(4*n, n)/(3*n + 1) for n >= 0.
%F A369082 (6.b) a(2*n+1) = 3*binomial(4*n+3,n)/(4*n+3) for n >= 0.
%e A369082 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 4*x^4 + 15*x^5 + 22*x^6 + 91*x^7 + 140*x^8 + 612*x^9 + 969*x^10 + 4389*x^11 + 7084*x^12 + ...
%e A369082 RELATED SERIES.
%e A369082 We can see from the expansion of A(x)^2, which begins
%e A369082 A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 15*x^4 + 44*x^5 + 91*x^6 + 280*x^7 + 612*x^8 + 1938*x^9 + 4389*x^10 + 14168*x^11 + 32890*x^12 + ...
%e A369082 that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
%e A369082 (A(x) - A(-x))/2 = x + 3*x^3 + 15*x^5 + 91*x^7 + 612*x^9 + ...
%e A369082 (A(x)^2 + A(-x)^2)/2 = 1 + 3*x^2 + 15*x^4 + 91*x^6 + 612*x^8 + ...
%e A369082 and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
%e A369082 (A(x) + A(-x))/2 = 1 + x^2 + 4*x^4 + 22*x^6 + 140*x^8 + 969*x^10 + ...
%e A369082 (A(x)^2 - A(-x)^2)/2 = 2*x + 8*x^3 + 44*x^5 + 280*x^7 + 1938*x^9 + ...
%e A369082 so that (A(x) + A(-x))/2 = 1 + (1/2)*x * (A(x)^2 - A(-x)^2)/2.
%o A369082 (PARI) {a(n) = if(n%2==0, binomial(2*n, n/2)/(3*n/2 + 1), 3*binomial(2*n+1,n\2)/(2*n+1))}
%o A369082 for(n=0,30, print1(a(n),", "))
%o A369082 (PARI) {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
%o A369082 A = 1 + x*(A^2 + B^2)/2 + (1/2)*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
%o A369082 for(n=0, 30, print1(a(n), ", "))
%o A369082 (PARI) {a(n) = my(A); A = (1/x)*serreverse( (sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) - (1 - 6*x + 4*x^2))/8 ); polcoeff(A, n)}
%o A369082 for(n=0, 30, print1(a(n), ", "))
%Y A369082 Cf. A369083, A368633, A368634, A368635, A368627, A368629.
%Y A369082 Cf. A000108, A001764, A002293, A006632.
%K A369082 nonn
%O A369082 0,4
%A A369082 _Paul D. Hanna_, Jan 13 2024