A391223 - OEIS

%I #25 Dec 05 2025 08:31:27

%S 1,0,3,12,48,192,769,3084,12381,49744,199977,804276,3235668,13020192,

%T 52400538,210909656,848950359,3417275616,13755595689,55369756356,

%U 222871001064,897051906816,3610447601694,14530522785576,58475788263354,235311984533472,946856171141034

%N Expansion of 1/(g * (2-g))^3, where g = 1+x*g^2 is the g.f. of A000108.

%H Seiichi Manyama, <a href="/A391223/b391223.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1 - x^2*g^4)^3, where g = 1+x*g^2 is the g.f. of A000108.

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

%F a(n) = (6/n) * Sum_{k=0..n-1} binomial(k+2,3) * binomial(2*n-4,n-1-k) for n > 0.

%F a(n) = (6/n) * Sum_{k=0..floor(n/2)} binomial(k+2,3) * binomial(2*n,n-2*k) for n > 0.

%F a(n) = 3*4^(n-2)/2 + binomial(2*n-2,n) + (n-1)*binomial(2*n-4,n-2)/2 for n > 1.

%F E.g.f.: (39 + 9*exp(4*x) - 36*x + 8*exp(2*x)*(2*(3 + (x - 6)*x)*BesselI(0, 2*x) + (13 - 2*x)*x*BesselI(1, 2*x)))/96. - _Stefano Spezia_, Dec 04 2025

%t Join[{1},Table[Sum[(6/n)*Binomial[k+2,3]*Binomial[2*n,n-2*k],{k,0,Floor[n/2]}],{n,1,25}]] (* _Vincenzo Librandi_, Dec 04 2025 *)

%o (PARI) a(n) = if(n<2, 0^n, 3*4^(n-2)/2+binomial(2*n-2, n)+(n-1)*binomial(2*n-4, n-2)/2);

%o (Magma) [1] cat [(6 / n) * &+[Binomial(k+2,3) * Binomial(2*n,n-2*k): k in [0..Floor(n/2)]] : n in [1..30] ]; // _Vincenzo Librandi_, Dec 04 2025

%Y Cf. A001791, A391222.

%Y Cf. A000108.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Dec 04 2025