A326809 G.f. A(x) = Sum_{n>=0} x^n * (A(x)^(n+1) + 1)^n / (1 + x*A(x)^n)^(n+1).

1, 1, 3, 12, 64, 388, 2547, 17675, 127930, 957361, 7363756, 57974777, 465801960, 3811089824, 31703423654, 267851394004, 2296630925851, 19975895528930, 176220976812512, 1576741746108772, 14312547251073466, 131857909192636473, 1233606830533043503, 11728329063674693906, 113406667874700311312, 1116271813812969589106, 11195131545541254173944

Offset

0,3

Links

Table of n, a(n) for n=0..26.

Formula

G.f. A(x) satisfies:

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

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

(3) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (A(x)^(n+1) - A(x)^k)^(n-k).

(4) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (A(x)^(n+1) + A(x)^k)^(n-k) * (-1)^k.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 388*x^5 + 2547*x^6 + 17675*x^7 + 127930*x^8 + 957361*x^9 + 7363756*x^10 + 57974777*x^11 + 465801960*x^12 + ...
satisfies
A(x) = 1/(1 - x*A(x)) + x*(A(x)^2 - 1)/(1 - x*A(x))^2 + x^2*(A(x)^3 - 1)^2/(1 - x*A(x)^2)^3 + x^3*(A(x)^4 - 1)^3/(1 - x*A(x)^3)^4 + x^4*(A(x)^5 - 1)^4/(1 - x*A(x)^4)^5 + x^5*(A(x)^6 - 1)^5/(1 - x*A(x)^5)^6 + ...
also
A(x) = 1/(1 + x*A(x)) + x*(A(x)^2 + 1)/(1 + x*A(x))^2 + x^2*(A(x)^3 + 1)^2/(1 + x*A(x)^2)^3 + x^3*(A(x)^4 + 1)^3/(1 + x*A(x)^3)^4 + x^4*(A(x)^5 + 1)^4/(1 + x*A(x)^4)^5 + x^5*(A(x)^6 + 1)^5/(1 + x*A(x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^(n+1) + 1)^n/(1 + x*Ser(A)^n)^(n+1) ), #A-1));
polcoeff(Ser(A), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^(n+1) - 1)^n/(1 - x*Ser(A)^n)^(n+1) ), #A-1));
polcoeff(Ser(A), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A324618.

Sequence in context: A233397 A206226 A371495 * A326557 A308204 A307724

Adjacent sequences: A326806 A326807 A326808 * A326810 A326811 A326812

Keyword

nonn

Author

Paul D. Hanna, Oct 19 2019

Status

approved