A391129 - OEIS

%I #25 Dec 03 2025 10:06:27

%S 1,2,8,36,177,920,4972,27656,157283,910378,5345408,31761232,190616428,

%T 1153815776,7035931524,43182501648,266538732951,1653488312726,

%U 10303782450616,64468708569236,404844870033361,2550760645474016,16120015360974736,102156302157189536

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

%H Vincenzo Librandi, <a href="/A391129/b391129.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g^3/(1 + x*g^2), where g = 1+x*g^3 is the g.f. of A001764.

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

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

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

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

%t Table[Sum[ (4*k+2)*Binomial[3*n -2*k+2,n-2*k]/(3*n-2*k+2),{k,0,Floor[n/2]}],{n,0,26}] (* _Vincenzo Librandi_, Nov 30 2025 *)

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

%o (Magma) [&+[(4*k+2)*Binomial(3*n-2*k+2, n-2*k)/(3*n-2*k+2): k in [0..Floor(n/2)]] : n in [0..40] ]; // _Vincenzo Librandi_, Nov 30 2025

%Y Cf. A109972, A391122, A391126.

%Y Cf. A089354, A121545.

%Y Cf. A001764, A006629.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 29 2025