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%I #15 Sep 06 2018 01:53:49
%S 1,1,4,21,122,758,4958,33509,233810,1641150,12364368,71807506,
%T 1354944972,-33794258600,2524565441138,-186642439700891,
%U 16196862324254354,-1602823227559245434,179707702260054046760,-22656977557634759678794,3191199098536326709613676,-499206960572108744520132444,86277300996554233583925645468,-16395890677314419248813441481150
%N G.f. A(x) satisfies: Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2.
%H Paul D. Hanna, <a href="/A277292/b277292.txt">Table of n, a(n) for n = 1..201</a>
%F G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
%F (1) A( A(x) + A(x)^2 ) = C(x),
%F (2) C( A(x) - A(x)^2 ) = A(x),
%F (3) A( A(x) - A(x)^2 ) = -C(-x),
%F (4) A( A(x-x^2) + A(x-x^2)^2 ) = x,
%F where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers, A000108.
%e G.f.: A(x) = x + x^3 + 4*x^5 + 21*x^7 + 122*x^9 + 758*x^11 + 4958*x^13 + 33509*x^15 + 233810*x^17 + 1641150*x^19 + 12364368*x^21 +...
%e such that Series_Reversion( A(x) + A(x)^2 ) = A(x) - A(x)^2, where
%e A(x)^2 = x^2 + 2*x^4 + 9*x^6 + 50*x^8 + 302*x^10 + 1928*x^12 + 12849*x^14 + 88122*x^16 + 621022*x^18 + 4411180*x^20 +...
%e A(x) + A(x)^2 = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 50*x^8 + 122*x^9 + 302*x^10 + 758*x^11 + 1928*x^12 + 4958*x^13 + 12849*x^14 + 33509*x^15 + 88122*x^16 + 233810*x^17 + 621022*x^18 + 1641150*x^19 + 4411180*x^20 +...
%e Also,
%e A( A(x) + A(x)^2 ) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 +...
%e which equals the Catalan series (A000108).
%o (PARI) {a(n) = my(Oxn=x*O(x^(2*n)), A = x +Oxn); for(i=1, 2*n, A = A + (x - subst(A+A^2, x, A-A^2 ))/2); polcoeff(A, 2*n-1)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A318008, A277293, A277294, A000108, A179270, A277292.
%K sign
%O 1,3
%A _Paul D. Hanna_, Oct 12 2016
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