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%I #21 Apr 02 2024 08:53:58
%S 1,2,12,98,924,9468,102432,1151410,13315692,157406876,1893480264,
%T 23103024084,285233168760,3556744196000,44730062281800,
%U 566683825859730,7225564521956940,92653105887920556,1194068058333608136,15457771628663418748,200916876963088849992
%N G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + A(x))^2/2.
%H G. C. Greubel, <a href="/A219538/b219538.txt">Table of n, a(n) for n = 0..870</a>
%F Let F(x) = (1-x - sqrt(1 - 4*x)) / x, then g.f. A(x) satisfies:
%F (1) A(x) = sqrt( (1/x)*Series_Reversion(x/F(x)^2) ),
%F (2) A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2),
%F where F(x) = 2*C(x) - 1 such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%F Let G(x) be the g.f. of A100327, then g.f. A(x) satisfies:
%F (3) A(x) = (1/x)*Series_Reversion(x/G(x)),
%F (4) A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
%F Recurrence: 3*n*(3*n-1)*(3*n+1)*(11*n-14)*a(n) = 3*(2*n-1)*(693*n^3 - 1575*n^2 + 1026*n - 176)*a(n-1) + 2*(n-2)*(2*n-3)*(2*n-1)*(11*n-3)*a(n-2). - _Vaclav Kotesovec_, Dec 28 2013
%F a(n) ~ sqrt(242+66*sqrt(33)) * (7+11/9*sqrt(33))^n / (66*sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 28 2013
%F a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} 2^(n-2*k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - _Seiichi Manyama_, Apr 02 2024
%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 98*x^3 + 924*x^4 + 9468*x^5 + 102432*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 4*x + 28*x^2 + 244*x^3 + 2384*x^4 + 24984*x^5 +...
%e A(2)^3 = 1 + 6*x + 48*x^2 + 446*x^3 + 4524*x^4 + 48588*x^5 +...
%e A(2)^4 = 1 + 8*x + 72*x^2 + 712*x^3 + 7504*x^4 + 82704*x^5 +...
%e where A(x) = 1 + x*(A(x)^2 + 2*A(x)^3 + A(x)^4)/2.
%e The g.f. satisfies A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2) where
%e F(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 +...+ 2*A000108(n)*x^n +...
%e The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
%e G(x) = 1 + 2*x + 8*x^2 + 42*x^3 + 252*x^4 + 1636*x^5 +...+ A100327(n)*x^n +...
%t CoefficientList[Sqrt[1/x*InverseSeries[Series[x^3/(1-x-Sqrt[1-4*x])^2, {x, 0, 20}], x]],x] (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o (PARI) /* Formula A(x) = 1 + x*A(x)^2*(1 + A(x))^2/2: */
%o {a(n)=local(A=1);for(i=1,n,A=1+x*A^2*(1+A +x*O(x^n))^2/2);polcoeff(A,n)}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) /* Formula using Series Reversion involving Catalan numbers: */
%o {a(n)=local(A=1);A=(1-x-sqrt(1-4*x +x^3*O(x^n)))/x; polcoeff(sqrt(1/x*serreverse(x/A^2)), n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A068875, A100327.
%Y Cf. A219534, A219537, A371658.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 22 2012
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