A391060 - OEIS

%I #13 Dec 01 2025 08:28:51

%S 1,1,4,15,68,333,1727,9317,51758,294068,1700827,9980653,59274015,

%T 355592894,2151707068,13117292347,80486635016,496688096931,

%U 3080658090710,19194089036648,120075994871807,753943787632022,4749729156662452,30013614335452595,190184864457166325,1208214303761406925

%N Expansion of g/(1 - x^2*g^2), where g = 1+x*g^3 is the g.f. of A001764.

%H Vincenzo Librandi, <a href="/A391060/b391060.txt">Table of n, a(n) for n = 0..1000</a>

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

%t Table[Sum[(2*k+1)*Binomial[3*n-4*k+1,n-2*k]/(3*n-4*k+1),{k,0,Floor[n/2]}],{n,0,25}] (* _Vincenzo Librandi_, Dec 01 2025 *)

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

%o (Magma) [&+[(2*k+1)*Binomial(3*n-4*k+1, n-2*k)/(3*n-4*k+1): k in [0..Floor(n/2)]] : n in [0..40] ]; // _Vincenzo Librandi_, Dec 01 2025

%Y Cf. A000305, A047099, A109957, A121545, A391058.

%Y Cf. A001764.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 26 2025