A364739 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^4).

1, 1, 3, 14, 78, 477, 3094, 20905, 145547, 1036891, 7522335, 55382045, 412740298, 3107671807, 23604165913, 180641336755, 1391555475647, 10781886600707, 83968131035849, 656931982467460, 5160714860765430, 40692065290732340, 321937030883130021

Offset: 0

Author

Seiichi Manyama, Aug 05 2023

Keywords

• nonn

Crossrefs

Cf. A001003, A001764, A219537, A364740.

Programs

Mathematica

nmax = 22; A[_] = 1;
Do[A[x_] = 1 + x*A[x]^2/(1-x*A[x]^4) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

PARI

a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(2*n+2*k, n-1-k))/n);

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k,n-1-k) for n > 0.

From Paul D. Hanna, Aug 19 2023: (Start)

A(x)^m = Sum_{n>=0} a(n,m) * x^n, where a(n,m) = (m/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k+m-1,n-1-k) for n > 0.

A(x) = exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k-1,n-1-k) for n >= 1.

G.f. A(x) satisfies the following formulas.

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

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

(3) A(x)^3 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^7).

(4) A(x)^4 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^8).

(5) A(x)^5 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^9).

(6) A(x)^6 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10).

(End)