A390184 - OEIS

%I #12 Nov 04 2025 07:12:58

%S 1,1,3,16,173,2016,28327,487264,9742329,221970016,5695274411,

%T 162587676384,5111837454517,175549536496816,6539363083186767,

%U 262652023656051376,11315735944857843953,520566507042182202816,25470084710826749361619,1320718835171918257741504

%N E.g.f. A(x) satisfies A(x) = exp( x/(1-x^3)^2 * A(x) ).

%H Vincenzo Librandi, <a href="/A390184/b390184.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-3*k-1) * binomial(2*n-5*k-1,k)/(n-3*k)!.

%F E.g.f.: exp( -LambertW(-x/(1-x^3)^2) ).

%t Table[n!*Sum[(n-3*k+1)^(n-3*k-1)*Binomial[2*n-5*k-1,k]/(n-3*k)!,{k,0,Floor[n/3]}],{n,0,25}] (* _Vincenzo Librandi_, Nov 03 2025 *)

%o (PARI) a(n) = n!*sum(k=0, n\3, (n-3*k+1)^(n-3*k-1)*binomial(2*n-5*k-1, k)/(n-3*k)!);

%o (Magma) [Factorial(n) * &+[(n-3*k+1)^(n-3*k-1)* Binomial(2*n-5*k-1, k) / Factorial(n-3*k) : k in [0..Floor(n/3)]] : n in [0..25] ]; // _Vincenzo Librandi_, Nov 03 2025

%Y Cf. A376576, A390014, A390015, A390185.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 28 2025