A188573 - OEIS

%I #24 Apr 11 2018 08:18:01

%S 0,0,2,6,32,120,528,2128,8960,36864,153472,635008,2635776,10922496,

%T 45300736,187800576,778731520,3228696576,13387309056,55506722816,

%U 230146834432,954246856704,3956565671936,16404954546176,68019305840640,282025965649920

%N Coefficient of the sqrt(6) term in (1 + sqrt(2) + sqrt(3))^n, denoted as C6(n).

%H G. C. Greubel, <a href="/A188573/b188573.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)

%F From _G. C. Greubel_, Apr 10 2018: (Start)

%F Empirical: a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) + 8*a(n-4).

%F Empirical: G.f.: 2*x^2*(1-x)/(1 - 4*x - 4*x^2 + 16*x^3 - 8*x^4). (End)

%e C6(3) is equal to 6, because (1+sqrt(2)+sqrt(3))^3 = 16 + 14 sqrt(2) + 12 sqrt(3) + 6 sqrt(6).

%t C6[n_] := Sum[Sum[2^(Floor[n/2] - j - 1 - k) 3^j Multinomial[2 k + n - 2 Floor[n/2], 2 j + 1, 2 Floor[n/2] - 2 k - 1 - 2 j], {j, 0, Floor[n/2] - k - 1}], {k, 0, Floor[n/2] - 1}]; Table[C6[n], {n, 0, 25}]

%t a[n_] := Coefficient[ Expand[(1 + Sqrt[2] + Sqrt[3])^n], Sqrt[6]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 08 2013 *)

%Y Cf. A188570, A188571, A188572.

%K nonn

%O 0,3

%A _Mateusz Szymański_, Dec 28 2012

%E Keyword tabl removed by _Michel Marcus_, Apr 11 2018