A392978 - OEIS

%I #16 Jan 31 2026 16:48:56

%S 1,1,2,7,26,98,387,1590,6699,28754,125359,553696,2472267,11140492,

%T 50599174,231401382,1064646228,4924458114,22886125061,106815213813,

%U 500447918795,2352844049280,11096870017710,52488362170335,248930058920517,1183456657164537,5639068382464958

%N Expansion of (1/x) * Series_Reversion( x * ((1-x^3)^2 - x) ).

%H Vincenzo Librandi, <a href="/A392978/b392978.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies A(x) = 1/((1 - (x*A(x))^3)^2 - x*A(x)).

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n-3*k,n) * binomial(4*n-5*k+1,k).

%t Table[1/(n+1)*Sum[Binomial[2*n-3*k,n]*Binomial[4*n-5*k+1,k],{k,0,Floor[n/3]}],{n,0,24}] (* _Vincenzo Librandi_, Jan 31 2026 *)

%o (PARI) a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)*binomial(4*n-5*k+1, k))/(n+1);

%o (Magma) [1/(n+1) * &+[Binomial(2*n-3*k, n)* Binomial(4*n-5*k+1, k) : k in [0..Floor(n/3)]] : n in [0..26] ]; // _Vincenzo Librandi_, Jan 31 2026

%Y Cf. A049140, A370840, A392979.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 29 2026