A100240 G.f. A(x) satisfies: 4^n/2 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: ((4+z)^n + z^n)/2 = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
1, 1, 2, 2, 0, -4, -6, 2, 22, 30, -26, -154, -172, 288, 1190, 990, -3040, -9620, -4970, 31350, 79120, 12580, -318210, -649610, 174150, 3185686, 5233514, -4273078, -31452228, -40495600, 64593386, 305819154, 290278982, -835918098, -2921409370, -1771072346, 9995237616, 27317409988
Offset: 0
Keywords
Examples
From the table of powers of A(x), we see that 4^n/2 = Sum of coefficients [x^0] through [x^n] in A(x)^n: A^1=[1,1],2,2,0,-4,-6,2,22,30,-26,... A^2=[1,2,5],8,8,0,-16,-24,8,88,120,... A^3=[1,3,9,19],30,30,2,-54,-84,20,288,... A^4=[1,4,14,36,73],112,112,16,-176,-288,32,... A^5=[1,5,20,60,145,281],420,420,90,-570,-988,... A^6=[1,6,27,92,255,582,1085],1584,1584,440,-1848,... A^7=[1,7,35,133,413,1071,2331,4201],6006,6006,2002,... A^8=[1,8,44,184,630,1816,4460,9320,16305],22880,22880,... the main diagonal of which is: [x^n]A(x)^(n+1) = (n+1)*A081696(n) for n>=0.
Programs
-
Mathematica
CoefficientList[Series[2*x + Sqrt[1 - 2*x + 5*x^2], {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 07 2021 *) -
PARI
a(n)=if(n==0,1,(4^n/2-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)
-
PARI
a(n)=polcoeff(2*x+sqrt(1-2*x+5*x^2+x^2*O(x^n)),n)
Formula
G.f.: A(x) = 2*x+sqrt(1-2*x+5*x^2).
Recurrence: n*a(n) = (2*n-3)*a(n-1) - 5*(n-3)*a(n-2). - Vaclav Kotesovec, Feb 07 2021