A391303 - OEIS

%I #13 Dec 07 2025 08:43:23

%S 1,2,7,30,142,722,3843,21136,119156,684886,3998441,23644650,141331431,

%T 852524958,5183033965,31726664930,195373188850,1209498757272,

%U 7523063932900,46991671366922,294647736792191,1853899154252454,11701336203814681,74068582235696580

%N Expansion of g^2/(1 + x^4*g^4), where g = 1+x*g^3 is the g.f. of A001764.

%H Vincenzo Librandi, <a href="/A391303/b391303.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/4)} (-1)^k * (4*k+2) * binomial(3*n-8*k+2,n-4*k)/(3*n-8*k+2).

%t Table[ Sum[(-1)^k*(4*k+2)*Binomial[3*n-8*k+2,n-4*k]/(3*n-8*k+2),{k,0,Floor[n/4]}],{n,0,21}] (* _Vincenzo Librandi_, Dec 07 2025 *)

%o (PARI) a(n) = sum(k=0, n\4, (-1)^k*(4*k+2)*binomial(3*n-8*k+2, n-4*k)/(3*n-8*k+2));

%o (Magma) [&+[(-1)^k*(4*k+2)*Binomial(3*n-8*k+2, n-4*k)/(3*n-8*k+2): k in [0..Floor(n/4)]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 07 2025

%Y Cf. A001764, A098746.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 06 2025