A352878 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (A(x) - x^n)^n.

1, 1, 1, 4, 9, 28, 86, 265, 855, 2783, 9214, 30867, 104481, 356859, 1228064, 4254760, 14827717, 51944178, 182819043, 646128479, 2292209747, 8159722774, 29137365046, 104343145880, 374640974686, 1348388811894, 4863897209014, 17581361532197, 63673157211566

Offset

0,4

Comments

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).

Links

Table of n, a(n) for n=0..28.

Formula

G.f. A(x) satisfies:

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

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

Example

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 + ...
where
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 + ...
also,
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 + ...
Specific values.
A(1/4) = 1.56854185594017483280286112046695218005165445150...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n,
A = sum(m=0, n, x^m * (A - x^m)^m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A352816, A317997.

Sequence in context: A094255 A192234 A069563 * A276984 A210969 A059345

Adjacent sequences: A352875 A352876 A352877 * A352879 A352880 A352881

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2022

Status

approved