OFFSET
0,3
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)
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 *)
PROG
(PARI) a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(2*n+2*k, n-1-k))/n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 05 2023
STATUS
approved