A216434 - OEIS

%I #23 Jan 21 2023 07:06:39

%S 1,2,4,10,27,74,208,600,1762,5244,15788,48006,147199,454618,1412960,

%T 4416016,13869998,43756124,138587784,440523892,1404849486,4493472836,

%U 14411800352,46338611632,149338703380,482315951104,1560824670460,5060345244766,16434480777703

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

%C The radius of convergence of the g.f. A(x) is r = 1/(2+sqrt(2)) with A(r) = 2*(1+sqrt(2)).

%C More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then

%C A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )

%C where the radius of convergence r of the g.f. A(x) satisfies

%C r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).

%H G. C. Greubel, <a href="/A216434/b216434.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: ((1-2*x) - sqrt( (1-2*x)^2 - 4*x^3*(2-x) )) / (2*x^3*(2-x)).

%F Recurrence: 2*(n+3)*a(n) = 3*(3*n+5)*a(n-1) - 6*(2*n+1)*a(n-2) + 4*(5*n-6)*a(n-3) - 4*(4*n-9)*a(n-4) + 4*(n-3)*a(n-5). - _Vaclav Kotesovec_, Aug 18 2013

%F a(n) ~ 2^(n/2 + 5/4) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Aug 18 2013, simplified Jan 21 2023

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 27*x^4 + 74*x^5 + 208*x^6 + 600*x^7 +...

%e The logarithm of the g.f. begins:

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

%e ((2-x)^3 + 3^2*x*(2-x)^2 + 3^2*x^2*(2-x) + x^3)*x^3/3 +

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

%e ((2-x)^5 + 5^2*x*(2-x)^4 + 10^2*x^2*(2-x)^3 + 10^2*x^3*(2-x)^2 + 5^2*x^4*(2-x) + x^5)*x^5/5 +...

%e Explicitly,

%e log(A(x)) = 2*x + 4*x^2/2 + 14*x^3/3 + 44*x^4/4 + 132*x^5/5 + 412*x^6/6 + 1318*x^7/7 + 4236*x^8/8 + 13676*x^9/9 + 44424*x^10/10 +...

%t CoefficientList[Series[((1 - 2*x) - Sqrt[(1 - 2*x)^2 - 4*x^3*(2 - x)])/(2*x^3*(2 - x)), {x,0,50}], x] (* _G. C. Greubel_, Feb 03 2017 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(2-x)^(m-k) + x*O(x^n)))),n)}

%o (PARI) {a(n)=polcoeff(2/(1-2*x+sqrt((1-2*x)^2-4*x^3*(2-x) +x*O(x^n))),n)}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A216604, A216454, A192415.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 10 2012