A336945 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.

1, -1, 1, 6, 31, 163, 882, 4896, 27759, 160149, 937365, 5553210, 33237828, 200696356, 1221105376, 7479222624, 46079243631, 285373035417, 1775569951995, 11093660204970, 69574265317095, 437832231422355, 2763889941603630, 17497374053053440, 111061430519553540, 706647507156148428, 4506221447451530172

Offset

0,4

Links

Table of n, a(n) for n=0..26.

Formula

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

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

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

Maple

# Recurrence:
a := proc(n) option remember; if n < 4 then return [1, -1, 1, 6][n+1] fi;
-((-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:
seq(a(n), n=0..26); # Peter Luschny, Aug 09 2020
alias(PS=ListTools:-PartialSums): A336945List := proc(m) local A, P, n;
A := [1, -1, 1]; P := [1, 1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A336945List(26); # Peter Luschny, Mar 26 2022

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A001764.

Sequence in context: A003128 A058146 A015449 * A162475 A343350 A036729

Adjacent sequences: A336942 A336943 A336944 * A336946 A336947 A336948

Keyword

sign

Author

Petros Hadjicostas, Aug 08 2020

Status

approved