A394495 - OEIS

%I #19 Mar 30 2026 23:59:25

%S 1,1,8,126,3000,96096,3877632,188972784,10801574784,708750000000,

%T 52514194876416,4337146638849024,395110823965304832,

%U 39360597010307629056,4256752500000000000000,496690565175807256590336,62197673632104828591341568,8320198233259787899473985536

%N Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - 5*x)^(1/5) ).

%H Vincenzo Librandi, <a href="/A394495/b394495.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) satisfies A(x) = 1/(1 - 5*x*A(x))^(1/5).

%F a(n) = 5^n * n! * binomial((6*n-4)/5,n)/(n+1).

%F a(n) ~ 6^(6*n/5-3/10)*exp(-n)*n^(n-1). - _Stefano Spezia_, Mar 22 2026

%t a[n_]:=5^n*n!*Binomial[(6 n-4)/5,n]/(n+1); Table[a[n],{n,0,20}] (* _Vincenzo Librandi_, Mar 30 2026 *)

%o (PARI) a(n) = 5^n*n!*binomial((6*n-4)/5, n)/(n+1);

%o (Magma) [1] cat [5^n*Factorial(n)*&*[((6*n-4)/5-k): k in [0..n-1]]/Factorial(n)/(n+1): n in [1..20]]; // _Vincenzo Librandi_, Mar 30 2026

%Y Cf. A000142, A001761, A282626, A385369, A385440, A393169, A394494.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Mar 22 2026