A392977 - OEIS

%I #16 Jan 31 2026 16:48:53

%S 1,1,5,20,101,525,2887,16359,95115,563838,3395642,20716254,127764434,

%T 795256210,4989327246,31518412352,200311909083,1279869928686,

%U 8216519744870,52973522292775,342843762187861,2226606216608385,14506570969312335,94785565057725600

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

%H Vincenzo Librandi, <a href="/A392977/b392977.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n-2*k,n) * binomial(6*n-5*k+2,k).

%t Table[1/(n+1)*Sum[Binomial[2*n-2*k,n]*Binomial[6*n-5*k+2,k],{k,0,Floor[n/2]}],{n,0,24}] (* _Vincenzo Librandi_, Jan 31 2026 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*n-2*k, n)*binomial(6*n-5*k+2, k))/(n+1);

%o (Magma) [1/(n+1) * &+[Binomial(2*n-2*k, n)* Binomial(6*n-5*k+2, k) : k in [0..Floor(n/2)]] : n in [0..25] ]; // _Vincenzo Librandi_, Jan 31 2026

%Y Cf. A001002, A370839, A392976.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 29 2026