A336945 - OEIS

%I #24 Mar 26 2022 04:24:03

%S 1,-1,1,6,31,163,882,4896,27759,160149,937365,5553210,33237828,

%T 200696356,1221105376,7479222624,46079243631,285373035417,

%U 1775569951995,11093660204970,69574265317095,437832231422355,2763889941603630,17497374053053440,111061430519553540,706647507156148428,4506221447451530172

%N a(n) = binomial(3*n,n)/(2*n + 1) - 2*binomial(3*(n - 1),n - 1)/(2*n - 1) for n > 0 with a(0) = 1.

%F a(n) = if(n == 0, 1, A001764(n) - 2*A001764(n-1)).

%F O.g.f.: (1 - 2*z)*2*sin(arcsin(sqrt(27*z)/2)/3)/sqrt(3*z). [This is due to _Emeric Deutsch_.]

%F D-finite with recurrence 2*n*(2*n+1)*a(n) +(-43*n^2+67*n-18)*a(n-1) +4*(31*n^2-130*n+117)*a(n-2) -12*(3*n-10)*(3*n-11)*a(n-3)=0. - _R. J. Mathar_, Mar 06 2022

%p # Recurrence:

%p a := proc(n) option remember; if n < 4 then return [1, -1, 1, 6][n+1] fi;

%p -((-108*n^2 + 756*n - 1320)*a(n - 3) + (124*n^2 - 520*n + 468)*a(n - 2) + (-43*n^2 + 67*n - 18)*a(n - 1)) / (4*n^2 + 2*n) end:

%p seq(a(n), n=0..26); # _Peter Luschny_, Aug 09 2020

%p alias(PS=ListTools:-PartialSums): A336945List := proc(m) local A, P, n;

%p A := [1,-1,1]; P := [1,1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));

%p A := [op(A), P[-1]] od; A end: A336945List(26); # _Peter Luschny_, Mar 26 2022

%t a[n_] := Binomial[3*n, n]/(2*n + 1) - 2 * Binomial[3*(n - 1), n - 1]/(2*n - 1); Array[a, 27, 0] (* _Amiram Eldar_, Aug 08 2020 *)

%o (PARI) a(n) = if (n!=0, binomial(3*n,n)/(2*n + 1) - 2*binomial(3*(n - 1),n - 1)/(2*n - 1), 1); \\ _Michel Marcus_, Aug 09 2020

%Y Cf. A001764.

%K sign

%O 0,4

%A _Petros Hadjicostas_, Aug 08 2020