A232732 - OEIS

%I #39 Jan 15 2022 23:26:24

%S -922,2434966,-6575675482,17767242206806,-48007067114436442,

%T 129715076631793674646,-350490088991612425827802,

%U 947024090736392816800774486,-2558858742679396890519761433562,6914035375695623821224314247122326,-18681721026270831871754901657845477722

%N a(n) = Sum_{k=0..2*n} (-1)^k * binomial(12*n,6*k).

%C a(2n) == 1 (mod 3); a(2n+1) == 2 (mod 3) Any proof?

%C This follows from the recurrence relation. - _Charles R Greathouse IV_, Dec 13 2013

%H Colin Barker, <a href="/A232732/b232732.txt">Table of n, a(n) for n = 1..290</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2766,-172929,-64).

%F a(n) = (-1)^n/3 * (2^(6*n) + ((sqrt(3)+1)^(12*n) + (sqrt(3)-1)^(12*n))/2^(6*n)). - _Vaclav Kotesovec_, Dec 06 2013

%F G.f.: -2*x*(32*x^2+57643*x+461) / ((64*x+1)*(x^2+2702*x+1)). - _Colin Barker_, Dec 06 2013

%F a(n) = -2766*a(n-1)-172929*a(n-2)-64*a(n-3). - _Wesley Ivan Hurt_, Jan 15 2022

%p A232732:=n->add((-1)^i*binomial(12*n,6*i), i=0..2*n); seq(A232732(n), n=1..20); # _Wesley Ivan Hurt_, Dec 06 2013

%t A[n_] := Sum[(-1)^k Binomial[12 n, 6 k], {k, 0, 2n}]; Array[A,14]

%t CoefficientList[Series[-2 (32 x^2 + 57643 x + 461) / ((64 x + 1) (x^2 + 2702 x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 09 2014 *)

%t LinearRecurrence[{-2766,-172929,-64},{-922,2434966,-6575675482},20] (* _Harvey P. Dale_, Apr 18 2019 *)

%o (PARI) a(n)=sum(k=0,2*n,(-1)^k*binomial(12*n,6*k)) \\ _Charles R Greathouse IV_, Dec 13 2013

%o (PARI) Vec(-2*x*(32*x^2+57643*x+461)/((64*x+1)*(x^2+2702*x+1)) + O(x^100)) \\ _Colin Barker_, Nov 09 2014

%Y Cf. A232719.

%K sign,easy

%O 1,1

%A _José María Grau Ribas_, Nov 29 2013