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A229042 Series reversion of (sqrt(1+4*x) - 1)/2 - x^2.

%I A229042 #25 Mar 24 2023 14:39:01
%S A229042 1,2,6,25,114,560,2880,15321,83600,465322,2631668,15079922,87362184,
%T A229042 510837760,3010987912,17870854206,106713713826,640659094566,
%U A229042 3864643224900,23412690485800,142386465217920,868967571732540,5320093500153120,32666064906596550,201109232686971492
%N A229042 Series reversion of (sqrt(1+4*x) - 1)/2 - x^2.
%H A229042 G. C. Greubel, <a href="/A229042/b229042.txt">Table of n, a(n) for n = 1..1000</a>
%F A229042 G.f. A(x) satisfies:
%F A229042 (1) A(x)^2 = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*(1+x)^(2*n)/n!.
%F A229042 (2) A(x)^2 = A(x)*C(-A(x)) - x, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F A229042 (3) A(x) = G(x)*(1 + G(x)) = sqrt(G(x) - x) where G(x) is the g.f. of A214372.
%F A229042 a(n) ~ r^(1/2-n) / (2*sqrt(Pi) * sqrt(1+8*s^3) * n^(3/2)), where s = ((3392 - 384*sqrt(78))^(1/3) + 4*(53 + 6*sqrt(78))^(1/3) - 4)/48 = 0.328649053... is the root of the equation 4*s^2*(1+4*s) = 1 and r = -1/2 + 1/(4*s) - s^2 = 0.152679653319... - _Vaclav Kotesovec_, Jan 22 2014
%F A229042 a(n) = binomial(2*(n-1),(n-1))/n-sum(k=1..n-1, k*(-1)^(k-n)*binomial(n+k-1,n-1)*((sum(l=0..k-1, (binomial(k-1,l)*binomial(2*(n+l-k),n-k-1))/(n+l-k)))))/n. - _Vladimir Kruchinin_, Feb 08 2015
%F A229042 D-finite with recurrence 3503*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(3377*n-3903)*a(n-1) -20*(n-2)*(2668*n^2-11692*n+12387)*a(n-2) +8*(-6616*n^3+57120*n^2-162464*n+152595)*a(n-3) -480*(4*n-15)*(4*n-13)*(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Mar 24 2023
%e A229042 G.f.: A(x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...
%e A229042 where the series reversion of g.f. A(x) begins:
%e A229042 (sqrt(1+4*x) - 1)/2 - x^2 = x - 2*x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +...+ (-1)^(n-1)*A000108(n-1)*x^n +...
%e A229042 The square of the g.f. equals the series:
%e A229042 A(x)^2 = x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! + d^4/dx^4 x^10*(1+x)^10/5! +...
%e A229042 A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...
%e A229042 Related Expansions:
%e A229042 G.f. A(x) = G(x) + G(x)^2 = sqrt(G(x) - x) where G(x) is the g.f. of A214372:
%e A229042 G(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...
%t A229042 Rest[CoefficientList[InverseSeries[Series[(Sqrt[1+4*x]-1)/2-x^2,{x,0,20}],x],x]] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o A229042 (PARI) {a(n)=polcoeff( serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^2 ), n)}
%o A229042 for(n=1, 30, print1(a(n), ", "))
%o A229042 (PARI) /* G.f. A(x) = sqrt(G(x) - x) where G(x) = x + G(x)^2*(1 + G(x))^2 */
%o A229042 {a(n)=local(G=serreverse(x-x^2*(1+x)^2+x^2*O(x^n)));polcoeff(sqrt(G-x),n)}
%o A229042 for(n=1, 30, print1(a(n), ", "))
%o A229042 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ n-th derivative
%o A229042 {a(n)=local(A2=x); A2=sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(sqrt(A2), n)}
%o A229042 for(n=1, 30, print1(a(n), ", "))
%o A229042 (Maxima)
%o A229042 a(n):=binomial(2*(n-1),(n-1))/n-sum(k*(-1)^(k-n)*binomial(n+k-1,n-1)*((sum((binomial(k-1,l)*binomial(2*(n+l-k),n-k-1))/(n+l-k),l,0,k-1))),k,1,n-1)/n; /* _Vladimir Kruchinin_, Feb 08 2015 */
%Y A229042 Cf. A214372, A229043.
%K A229042 nonn
%O A229042 1,2
%A A229042 _Paul D. Hanna_, Oct 24 2013