A324618 - OEIS

A324618

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

1, 1, 2, 5, 19, 86, 436, 2378, 13731, 83077, 523275, 3416329, 23051600, 160440679, 1150435934, 8492238919, 64508971958, 504172573079, 4053925852485, 33535370139607, 285391912938870, 2498255748837089, 22489737035242848, 208124346717364948, 1978949027666465869, 19321957528006663637, 193581292284734286398, 1988536950750112238165

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OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

G.f. A(x) satisfies:

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

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

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

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 86*x^5 + 436*x^6 + 2378*x^7 + 13731*x^8 + 83077*x^9 + 523275*x^10 + 3416329*x^11 + 23051600*x^12 + ...

such that

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

also

A(x) = 1/(1 + x) + x*(A(x) + 1)/(1 + x*A(x))^2 + x^2*(A(x)^2 + 1)^2/(1 + x*A(x)^2)^3 + x^3*(A(x)^3 + 1)^3/(1 + x*A(x)^3)^4 + x^4*(A(x)^4 + 1)^4/(1 + x*A(x)^4)^5 + x^5*(A(x)^5 + 1)^5/(1 + x*A(x)^5)^6 + ...

MATHEMATICA

m = 35; A[_] = 0; Unprotect[Power]; 0^0 = 1; Protect[Power];

Do[A[x_] = Sum[ x^n (A[x]^n - 1)^n/(1 - x A[x]^n)^(n + 1), {n, 0, k}] + O[x]^k, {k, m}];

CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2019 *)

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)^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)^n/(1 + x*Ser(A)^n)^(n+1) ), #A-1));

polcoeff(Ser(A), n)}