OFFSET
0,2
FORMULA
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349310.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+3*k-2,n-1) for n > 0.
D-finite with recurrence 3*n*(52*n-187)*(3*n-1) *(3*n-2)*a(n) +(14392*n^4 -70190*n^3 +56951*n^2 +50237*n -49500)*a(n-1) +3*(-17252*n^4 +205959*n^3 -851664*n^2 +1432459*n -815652)*a(n-2) +18*(-472*n^4 +1294*n^3 +36359*n^2 -226731*n +361171)*a(n-3) -27*(n-5)*(404*n^3 -2235*n^2 -4058*n +26406)*a(n-4) -81*(n-5)*(n-6) *(8*n^2+358*n-1785)*a(n-5) +243*(n-5)*(n-6) *(n-7)*(4*n-31)*a(n-6)=0. - R. J. Mathar, Jul 25 2023
MAPLE
A364407 := proc(n)
if n = 0 then
1;
else
(-1)^(n-1)*add( binomial(n, k) * binomial(n+3*k-2, n-1), k=0..n)/n ;
end if;
end proc:
seq(A364407(n), n=0..70); # R. J. Mathar, Jul 25 2023
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^3) + 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, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(n+3*k-2, n-1))/n);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jul 23 2023
STATUS
approved