OFFSET
0,6
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) satisfies A(x) = 1/(1 - (exp((x*A(x))^4) - 1)^2/(x*A(x))^3).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (2*(n-4*k))!/(n-4*k)! * (2*n-4*k)! * Stirling2(n-3*k,2*(n-4*k))/(n-3*k)!.
MATHEMATICA
nmax = 25; CoefficientList[1/x * InverseSeries[Series[x - (E^(x^4) - 1)^2/x^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 23 2026 *)
Table[(1/(n+1))*Sum[(2*(n-4*k))!/(n-4*k)!*(2*n-4*k)!*Abs[StirlingS2[n-3*k, 2*(n-4*k)]/(n-3*k)!], {k, 0, Floor[n/4]}], {n, 0, 23}] (* Vincenzo Librandi, Jan 24 2026 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x-(exp(x^4)-1)^2/x^2)/x))
(Magma) [1/(n+1)* &+[Factorial(2*(n-4*k))/Factorial(n-4*k) * Factorial(2*n-4*k)*StirlingSecond(n-3*k, 2*(n-4*k))/Factorial(n-3*k) : k in [0..Floor(n/4)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 24 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 23 2026
STATUS
approved
