A391408 - OEIS

%I #7 Dec 09 2025 08:17:05

%S 1,3,11,38,133,468,1660,5932,21346,77301,281545,1030778,3791597,

%T 14006456,51941576,193301120,721697246,2702472854,10147300982,

%U 38197179692,144119279906,544941540968,2064663329656,7837201508508,29801069731228,113504582390857,432973857932389

%N Expansion of g^5/(1 + x*g)^2, where g = 1+x*g^2 is the g.f. of A000108.

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

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

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

%Y Cf. A065601, A114495, A389115.

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 08 2025