A245734 - OEIS

%I #16 Sep 08 2022 08:46:09

%S 0,1,2,6,20,74,294,1228,5318,23662,107512,496726,2326462,11020424,

%T 52706138,254148326,1234240140,6031310162,29635011990,146323849876,

%U 725635937678,3612656833694,18049975590512,90474958563374,454841633027198,2292796383312656

%N G.f. A(x) satisfies 0 = A(0) and 0 = f(x, A(x)) where f(u, v) = (v - u) * (1 + u*v) - v * (v + u).

%H G. C. Greubel, <a href="/A245734/b245734.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1 - x - x^2 - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4)) / (2 * (1 - x)).

%F G.f.: x / (1 - x - x^2 - (1 - x) / (1 - x - x^2 - (1 - x) / ...)) continued fraction.

%F a(n) = A245735(-n) for all n in Z.

%F 0 = a(n)*(n-1) + a(n+1)*(n+2) + a(n+2)*(n+5) + a(n+3)*(-9*n-27) + a(n+4)*(7*n+26) + a(n+5)*(-n-5) for all n in Z.

%F 0 = a(n)*(+a(n+1) +4*a(n+2) +7*a(n+3) -45*a(n+4) +40*a(n+5) -7*a(n+6)) + a(n+1)*(-2*a(n+1) -4*a(n+2) +31*a(n+3) -44*a(n+4) +24*a(n+5) -4*a(n+6)) + a(n+2)*(-2*a(n+2) +a(n+3) +4*a(n+4) -a(n+6)) +a(n+3)*(-27*a(n+3) +97*a(n+4) -109*a(n+5) +27*a(n+6)) +a(n+4)*(-18*a(n+4) +40*a(n+5) -16*a(n+6)) +a(n+5)*(+2*a(n+5) +a(n+6)) for all n in Z

%e G.f. = x + 2*x^2 + 6*x^3 + 20*x^4 + 74*x^5 + 294*x^6 + 1228*x^7 + 5318*x^8 + ...

%t CoefficientList[Series[(1-x-x^2 -Sqrt[1-6*x+3*x^2+2*x^3+x^4])/(2*(1-x)), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 06 2018 *)

%o (PARI) {a(n) = my(A); n++; A = O(x); if( n<0, for(k=0, -n/2, A = x / (1 + (x - x^2) + (x - x^2) * A)), for(k=1, n, A = x / (1 - (x + x^2) - (1 - x) * A));); polcoeff(A, abs(n)) };

%o (PARI) {a(n) = polcoeff( if( n<0, ((-1 - x + x^2) + sqrt(1 + 2*x + 3*x^2 - 6*x^3 + x^4 + x^2 * O(x^-n) )) / (2 * (x - x^2)), ((1 - x - x^2) - sqrt(1 - 6*x + 3*x^2 + 2*x^3 + x^4 + x * O(x^n) )) / (2 * (1 - x))), abs(n))};

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-x^2 -Sqrt(1-6*x+3*x^2+2*x^3+x^4))/(2*(1-x)))); // _G. C. Greubel_, Aug 06 2018

%Y Cf. A245735.

%K nonn

%O 0,3

%A _Michael Somos_, Jul 30 2014