OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..250
FORMULA
G.f.: (1/4) * Sum_{k>=0} Product_{j=0..k-1} (3/4 + j*x).
a(0) = 1; a(n) = -4*a(n-1) + 8*Sum_{k=0..n-1} binomial(n+1,k+1) * a(k) * a(n-1-k).
a(n) = (4/3)^n * A390902(n+1).
a(n) ~ 2^(2*n-2) * n^n / (3^n * exp(n) * (1/3 - log(4/3))^(n + 1/2)). - Vaclav Kotesovec, Jan 19 2026
E.g.f.: 1/4 - 1/(4 + 4*LambertW(-1, -4*exp(4*(-1 + x)/3)/3)). - Vaclav Kotesovec, Jan 20 2026
MATHEMATICA
numTerms=15; v={1}; Do[ v=Append[v, -4 v[[-1]] +8 Sum[Binomial[n+1, k+1] v[[k+1]] v[[n-k]], {k, 0, n-1}]], {n, numTerms-1}]; v (* Vincenzo Librandi, Jan 17 2026 *)
nmax = 20; Assuming[{x > 0}, CoefficientList[Series[1/4 - 1/(4 + 4*LambertW[-1, -4*E^(4*(-1 + x)/3)/3]), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jan 20 2026 *)
PROG
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-4*v[i]+8*sum(j=0, i-1, binomial(i+1, j+1)*v[j+1]*v[i-j])); v;
(Magma) N := 15; v := [1]; for n in [1..N-1] do Append(~v, -4*v[n] + 8*&+[Binomial(n+1, k+1)*v[k+1]*v[n-k] : k in [0..n-1]]); end for; v; // Vincenzo Librandi, Jan 17 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 23 2025
STATUS
approved
