A352878 - OEIS

%I #9 Jun 01 2022 17:41:26

%S 1,1,1,4,9,28,86,265,855,2783,9214,30867,104481,356859,1228064,

%T 4254760,14827717,51944178,182819043,646128479,2292209747,8159722774,

%U 29137365046,104343145880,374640974686,1348388811894,4863897209014,17581361532197,63673157211566

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (A(x) - x^n)^n.

%C Compare the g.f. to: C(x) = Sum_{n>=0} x^n * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%F G.f. A(x) satisfies:

%F (1) A(x) = Sum_{n>=0} x^n * (A(x) - x^n)^n.

%F (2) A(x) = Sum_{n>=0} (-1)^n * x^(n*(n+1)) / (1 - x^(n+1)*A(x))^(n+1).

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 28*x^5 + 86*x^6 + 265*x^7 + 855*x^8 + 2783*x^9 + 9214*x^10 + 30867*x^11 + 104481*x^12 + ...

%e where

%e A(x) = 1 + x*(A(x) - x) + x^2*(A(x) - x^2)^2 + x^3*(A(x) - x^3)^3 + x^4*(A(x) - x^4)^4 + x^5*(A(x) - x^5)^5 + x^6*(A(x) - x^6)^6 + ...

%e also,

%e A(x) = 1/(1 - x*A(x)) - x^2/(1 - x^2*A(x))^2 + x^6/(1 - x^3*A(x))^3 - x^12/(1 - x^4*A(x))^4 + x^20/(1 - x^5*A(x))^5 + ...

%e Specific values.

%e A(1/4) = 1.56854185594017483280286112046695218005165445150...

%o (PARI) {a(n) = my(A=1); for(i=1,n,

%o A = sum(m=0,n, x^m * (A - x^m)^m +x*O(x^n)));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = polcoeff( sum(m=0,sqrtint(#A)+1, (-1)^m * x^(m*(m+1))/(1 - x^(m+1)*Ser(A))^(m+1) ),#A-1));H=A;A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A352816, A317997.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Apr 06 2022