A228966 - OEIS

%I #35 Oct 14 2019 11:42:41

%S 1,4,26,208,1858,17764,177842,1840672,19536546,211483556,2325884778,

%T 25915289008,291914007042,3318712168516,38030817789154,

%U 438833757057344,5094403323613762,59458030569218756,697263250712144058,8211774425030092944,97084082739501794626

%N G.f. satisfies: A(x) = (1 + x*A(x))^2 * (1 + A(x)) / 2.

%H G. C. Greubel, <a href="/A228966/b228966.txt">Table of n, a(n) for n = 0..900</a>

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%F G.f. A(x) satisfies:

%F (1) A(x) = exp( x*(A(x) + A(x)^2) + Integral(A(x) + A(x)^2 dx) ).

%F (2) A(x) = (1/x)*Series_Reversion( x*(1-2*x-x^2)/(1+x)^2 ).

%F (3) A(x) = 1 + x*A(x)*(1 + A(x))*(2 + x*A(x)).

%F (4) A(x) = 1 + Sum_{n>=2} (-1)^n * 2*n * x^(n-1) * A(x)^n.

%F Recurrence: 2*n*(n+1)*(29*n-39)*a(n) = n*(754*n^2 - 1391*n + 519)*a(n-1) - (203*n^3 - 679*n^2 + 660*n - 180)*a(n-2) + (n-3)*(2*n-3)*(29*n-10)*a(n-3). - _Vaclav Kotesovec_, Dec 21 2013

%F a(n) ~ 1/174*sqrt(87)*sqrt((8946558 + 86826*sqrt(87))^(1/3)*((8946558 + 86826*sqrt(87))^(2/3) + 42978 + 174*(8946558 + 86826*sqrt(87))^(1/3))) / ((8946558 + 86826*sqrt(87))^(1/3) * sqrt(Pi)) * 6^(-n)*((16046 + 174*sqrt(87))^(2/3) + 634 + 26*(16046 + 174*sqrt(87))^(1/3))^n*(16046 + 174*sqrt(87))^(-n/3) * (1/n)^(3/2). - _Vaclav Kotesovec_, Dec 21 2013

%e G.f.: A(x) = 1 + 4*x + 26*x^2 + 208*x^3 + 1858*x^4 + 17764*x^5 +...

%e Related expansions.

%e (1 + x*A(x))^2 = 1 + 2*x + 9*x^2 + 60*x^3 + 484*x^4 + 4340*x^5 +...

%e (1 + A(x))/2 = 1 + 2*x + 13*x^2 + 104*x^3 + 929*x^4 + 8882*x^5 +...

%e A(x) + A(x)^2 = 2 + 12*x + 94*x^2 + 832*x^3 + 7914*x^4 + 78972*x^5 +...

%e log(A(x)) = 4*x + 36*x^2/2 + 376*x^3/3 + 4160*x^4/4 + 47484*x^5/5 +...

%t CoefficientList[InverseSeries[Series[x*(1-2*x-x^2)/(1+x)^2, {x, 0, 20}], x]/x,x] (* _Vaclav Kotesovec_, Dec 21 2013 *)

%o (PARI) {a(n)=polcoeff((serreverse(x*(1-2*x-x^2)/(1+x)^2 +x^2*O(x^n))/x),n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1);for(i=1,n,A=exp(x*(A+A^2)+intformal(A+A^2 +x*O(x^n))));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1);for(i=1,n,A=1+x*A*(1+A)*(2+x*A) +x*O(x^n));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A231552, A231553, A231554, A231556, A231615, A231616, A231618.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 10 2013