A211768 G.f. satisfies: A(x) = 1 + x*A(x/A(-x)).

1, 1, 1, 2, 4, 12, 35, 133, 497, 2256, 10123, 53131, 276210, 1638039, 9639943, 63526677, 416194299, 3009639922, 21672348693, 170290649517, 1334332599748, 11302630861664, 95587196023618, 867197921850406, 7862652321850812, 75983785567389333, 734442008292947615

Offset: 0

Author

Paul D. Hanna, Jun 06 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 35*x^6 + 133*x^7 +...
Related expansion:
x/A(-x) = x + x^2 + x^4 - x^5 + 6*x^6 - 14*x^7 + 72*x^8 - 250*x^9 + 1338*x^10 +..
		

Links

• Robert Israel, Table of n, a(n) for n = 0..220

Programs

Maple

eq:= 1 - A(x) + x*A(x/A(-x)):
AA[0]:= 1: c[0]:= 1:
for n from 1 to 50 do
  S:= series(eval(eq, A = unapply(AA[n-1]+c[n]*x^n, x)), x, n+1);
  c[n]:= solve(convert(S,polynom),c[n]);
  AA[n]:= AA[n-1]+c[n]*x^n;
od:
seq(c[n],n=0..50); # Robert Israel, Aug 02 2017

Mathematica

nmax = 26; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*A[x/A[-x]]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k];
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*subst(A,x,x/subst(A,x,-x+x*O(x^n)))); polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))