A302101 G.f. A(x) satisfies: A(x) = 1 + x * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).

1, 1, 3, 12, 57, 318, 2190, 20298, 265557, 4683954, 102511182, 2622117720, 75857421522, 2436766744728, 85885843375992, 3292322691248310, 136306472824555725, 6059583761048830290, 287832499902612461910, 14546503423112218184280, 779260996651844076919230, 44104811071523533909346100, 2629647430438623322437523380, 164732335319839237173383723220

Offset: 0

Author

Paul D. Hanna, Apr 07 2018

Keywords

• nonn

Comments

Compare to: C(x) = 1 + x*C(x) * (C(x) + 2*x*C'(x)) / (C(x) + x*C'(x)) holds when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

If G(x) = 1 + x * (G(x) + y*x*G'(x)) / (G(x) + x*G'(x)), then negative coefficients are present in the power series expansion of G(x) when y < t where constant t = 3.79622607101172329768286422639804088884917373644497484011451904053377050...

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 318*x^5 + 2190*x^6 + 20298*x^7 + 265557*x^8 + 4683954*x^9 + 102511182*x^10 + ...
such that A(x) = 1 + x * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..382

Crossrefs

Cf. A302102.

Programs

Maple

G:= 1:
for n from 1 to 30 do
  G:= convert(series(1+x*(G+4*x*diff(G,x))/(G +x*diff(G,x)),x,n+1),polynom);
od:
seq(coeff(G,x,n),n=0..30); # Robert Israel, Apr 08 2018

Mathematica

nmax = 23; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*(A[x] + 4*x*A'[x])/(A[x] + x*A'[x])) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

PARI

/* Differential Equation */
{a(n) = my(A=1); for(i=0,n, A = 1 + x*(A + 4*x*A')/(A + x*A' +x^2*O(x^n))); polcoeff(A,n)}
for(n=0, 30, print1(a(n),", "))

Formula

G.f. A(x) satisfies: A'(x) = A(x) * (A(x) - 1 - x) / (x*(1 + 4*x - A(x))).

G.f.: A(x) = exp( Integral (A(x) - 1 - x) / (x*(1 + 4*x - A(x))) dx ).

a(n) ~ n! * c * 3^n / n^2, where c = 0.030801423287090143490227542092... - Vaclav Kotesovec, Aug 31 2024