%I #41 Feb 21 2020 06:59:06

%S 1,1,1,2,5,11,23,51,120,286,681,1636,3985,9803,24257,60338,150931,

%T 379501,958360,2429294,6179380,15769380,40361087,103579221,266471500,

%U 687098810,1775440421,4596689688,11922774513,30977768907,80615085087,210103228155,548352756656,1433053608502

%N G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^3*A(x)^2).

%H Vincenzo Librandi, <a href="/A192415/b192415.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(2*k) * A(x)^k ).

%F D-finite with recurrence: 4*(n+1)*(n+2)*(217*n^3 - 1239*n^2 + 1838*n - 336)*a(n) = 6*(n+1)*(434*n^4 - 2261*n^3 + 2339*n^2 + 1792*n - 1344)*a(n-1) - (n-1)*(2821*n^4 - 13286*n^3 + 7829*n^2 + 18464*n - 4032)*a(n-2) + 6*(868*n^5 - 6258*n^4 + 13981*n^3 - 7438*n^2 - 7769*n + 6136)*a(n-3) + 2*(n-3)*(2*n - 5)*(434*n^3 - 1393*n^2 + 211*n + 1048)*a(n-4) + 2*(n-4)*(2*n - 7)*(217*n^3 - 588*n^2 + 11*n + 480)*a(n-5). - _Vaclav Kotesovec_, Sep 19 2013

%F a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 2.730683387097269698... is the root of the equation -4 - 8*d - 24*d^2 + 13*d^3 - 12*d^4 + 4*d^5 = 0 and c = 2.078548317061344694159945441842754... is the root of the equation -1 - 67*c^2 - 19811*c^4 + 36463*c^6 - 41664*c^8 + 7936*c^10 = 0. - _Vaclav Kotesovec_, Sep 19 2013, updated Nov 28 2016

%F a(n) = Sum_{k=0..n/2+1} C(n-k+2,k-1)*C(n-k+2,2*k-1)/(n-k+2). - _Vladimir Kruchinin_, Feb 12 2019

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 51*x^7 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 36*x^5 + 82*x^6 + 190*x^7 +...

%e A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 205*x^6 + 498*x^7 +...

%e where A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^4*A(x)^3.

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + x^2*A(x))*x + (1 + 2^2*x^2*A(x) + x^4*A(x)^2)*x^2/2 +

%e (1 + 3^2*x^2*A(x) + 3^2*x^4*A(x)^2 + x^6*A(x)^3)*x^3/3 +

%e (1 + 4^2*x^2*A(x) + 6^2*x^4*A(x)^2 + 4^2*x^6*A(x)^3 + x^8*A(x)^4)*x^4/4 +

%e (1 + 5^2*x^2*A(x) + 10^2*x^4*A(x)^2 + 10^2*x^6*A(x)^3 + 5^2*x^8*A(x)^4 + x^10*A(x)^5)*x^5/5 +...

%e Explicitly,

%e log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 31*x^5/5 + 70*x^6/6 + 176*x^7/7 + 469*x^8/8 + 1228*x^9/9 + 3161*x^10/10 +...

%t nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x^3*AGF^2) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* _Vaclav Kotesovec_, Sep 19 2013 *)

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^3*(A+x*O(x^n))^2)); polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^(2*j)*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

%o (PARI)

%o x='x; y='y; Fxy = (1+x*y) * (1 + x^3*y^2) - y;

%o seq(N) = {

%o my(y0 = 1 + O('x^N), y1=0);

%o for (k = 1, N,

%o y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy,y), y, y0);

%o if (y1 == y0, break()); y0 = y1);

%o Vec(y0);

%o };

%o seq(34) \\ _Gheorghe Coserea_, Nov 30 2016

%Y Cf. A198951, A198953, A198957, A007863, A036765, A104545.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 02 2011