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A215661 G.f. satisfies A(x) = (1 + 2*x*A(x)) * (1 + x*A(x)^2).

%I A215661 #23 Sep 08 2024 10:02:31
%S A215661 1,3,14,83,554,3966,29756,230915,1838162,14926346,123157572,
%T A215661 1029590062,8702171620,74238432924,638408311800,5528154378467,
%U A215661 48161687414498,421848099386322,3712675503776372,32815429463428794,291169073934720940,2592569269501484836
%N A215661 G.f. satisfies A(x) = (1 + 2*x*A(x)) * (1 + x*A(x)^2).
%C A215661 The radius of convergence of g.f. A(x) is r = 0.10464695509817751113707000... with A(r) = 2.224485325158190991256253303513498621559794760... where y = A(r) satisfies 9*y^3 - 22*y^2 + 8*y - 8 = 0.
%C A215661 r = 1/((2/27*((76276+3348*sqrt(93))^(2/3) + 1684 + 46*(76276 + 3348*sqrt(93))^(1/3))/(76276 + 3348*sqrt(93))^(1/3))). - _Vaclav Kotesovec_, Sep 16 2013
%H A215661 G. C. Greubel, <a href="/A215661/b215661.txt">Table of n, a(n) for n = 0..1000</a>
%H A215661 Paul Barry, <a href="https://arxiv.org/abs/1912.01126">Riordan arrays, the A-matrix, and Somos 4 sequences</a>, arXiv:1912.01126 [math.CO], 2019.
%F A215661 G.f. A(x) satisfies:
%F A215661 (1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * 2^k/A(x)^k ).
%F A215661 (2) A(x) = (1/x) * Series_Reversion( x*(1-x-2*x^2)/(1+2*x) ).
%F A215661 (3) A(x) = Sum_{n>=0} A001045(n+2) * x^n * A(x)^n, where A001045 is the Jacobsthal numbers.
%F A215661 The formal inverse of the g.f. A(x) is (sqrt(4-4*x+9*x^2) - (2+x))/(4*x^2).
%F A215661 a(n) = [x^n] ( (1+2*x)/(1-x-2*x^2) )^(n+1) / (n+1).
%F A215661 Recurrence: 9*n*(n+1)*(31*n-55)*a(n) = 2*n*(1426*n^2 - 3243*n + 1211)*a(n-1) - 8*(248*n^3 - 936*n^2 + 991*n - 240)*a(n-2) + 32*(n-2)*(2*n-5)*(31*n-24)*a(n-3). - _Vaclav Kotesovec_, Sep 16 2013
%F A215661 a(n) ~ 1/558*sqrt(186)*sqrt((556299836 + 9879948*sqrt(93))^(1/3) * ((556299836 + 9879948*sqrt(93))^(2/3) + 669724 + 806*(556299836 + 9879948*sqrt(93))^(1/3)))/((556299836 + 9879948*sqrt(93))^(1/3)) * (2/27*((76276+3348*sqrt(93))^(2/3) + 1684 + 46*(76276+3348*sqrt(93))^(1/3))/ (76276+3348*sqrt(93))^(1/3))^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 16 2013
%F A215661 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - _Seiichi Manyama_, Sep 08 2024
%e A215661 G.f.: A(x) = 1 + 3*x + 14*x^2 + 83*x^3 + 554*x^4 + 3966*x^5 + 29756*x^6 + ...
%e A215661 Related expansions.
%e A215661 A(x)^2 = 1 + 6*x + 37*x^2 + 250*x^3 + 1802*x^4 + 13580*x^5 + 105709*x^6 + ...
%e A215661 A(x)^3 = 1 + 9*x + 69*x^2 + 528*x^3 + 4122*x^4 + 32847*x^5 + ...
%e A215661 where A(x) = 1 + x*(2*A(x) + A(x)^2) + 2*x^2*A(x)^3.
%e A215661 The g.f. also satisfies the series:
%e A215661 A(x) = 1 + 3*x*A(x) + 5*x^2*A(x)^2 + 11*x^3*A(x)^3 + 21*x^4*A(x)^4 + 43*x^5*A(x)^5 + 85*x^6*A(x)^6 + ... + Jacobsthal(n+2)*x^n*A(x)^n + ...
%e A215661 The logarithm of the g.f. equals the series:
%e A215661 log(A(x)) = (1 + 2/A(x))*x*A(x) + (1 + 2^2*2/A(x) + 2^2/A(x)^2)*x^2*A(x)^2/2 +
%e A215661 (1 + 3^2*2/A(x) + 3^2*2^2/A(x)^2 + 2^3/A(x)^3)*x^3*A(x)^3/3 +
%e A215661 (1 + 4^2*2/A(x) + 6^2*2^2/A(x)^2 + 4^2*2^3/A(x)^3 + 2^4/A(x)^4)*x^4*A(x)^4/4 +
%e A215661 (1 + 5^2*2/A(x) + 10^2*2^2/A(x)^2 + 10^2*2^3/A(x)^3 + 5^2*2^4/A(x)^4 + 2^5/A(x)^5)*x^5*A(x)^5/5 + ...
%e A215661 Explicitly,
%e A215661 log(A(x)) = 3*x + 19*x^2/2 + 150*x^3/3 + 1251*x^4/4 + 10738*x^5/5 + 93934*x^6/6 + 832716*x^7/7 + 7454755*x^8/8 + ... + L(n)*x^n/n + ...
%e A215661 where L(n) = [x^n] (1+2*x)^n/(1-x-2*x^2)^n.
%t A215661 CoefficientList[1/x * InverseSeries[Series[x*(1-x-2*x^2)/(1+2*x),{x,0,20}],x],x] (* _Vaclav Kotesovec_, Sep 16 2013 *)
%o A215661 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + 2*x*A)*(1 + x*(A+x*O(x^n))^2)); polcoeff(A, n)}
%o A215661 (PARI) {a(n)=polcoeff( (1/x)*serreverse( x*(1-x-2*x^2)/(1+2*x +x*O(x^n))), n)}
%o A215661 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*2^j/A^j)*x^m*A^m/m))); polcoeff(A, n)}
%o A215661 for(n=0,31,print1(a(n),", "))
%Y A215661 Cf. A215654, A216314, A001045.
%K A215661 nonn
%O A215661 0,2
%A A215661 _Paul D. Hanna_, Aug 19 2012