A268538 a(n) is the n-th prime 3-dimensional Catalan number.

1, 1, 2, 12, 107, 1178, 14805, 203885, 3002973, 46573347, 752521980, 12571607865, 215925120675, 3796546970232, 68106673339365, 1243210765414512, 23041656826384341, 432836397965356500, 8228713546454487768, 158123766469263772336, 3068030287437565705610, 60051703950609983469312

Offset

0,3

Comments

"Prime" here is used in the sense of "primitive" or "irreducible".

Links

Table of n, a(n) for n=0..21.

Manuel Wettstein, Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers, arXiv:1602.07235 [cs.CG], 2016.

Formula

Lemma 15 of Wettstein (2016) gives a formula in terms of the 3-dimensional Catalan numbers (A005789).

Maple

A005789 := proc(n)
    2*(3*n)!/(n+2)!/(n+1)!/n! ;
end proc:
maxn := 30 :
Cx := add(A005789(i)*x^i, i=0..maxn) ;
d := 3:
for i from 0 to maxn do
    coeftayl(1/Cx^(d*i-1), x=0, i) ;
    %/(1-d*i) ;
    printf("%d, ", %) ;
end do: # R. J. Mathar, Feb 27 2018

Mathematica

A005789[n_] := 2*(3*n)!/(n+2)!/(n+1)!/n!; Maxn = 30; Cx = Sum[A005789[i]* x^i, {i, 0, Maxn}]; d = 3; Reap[For[i = 0, i <= Maxn, i++, sc = SeriesCoefficient[1/Cx^(d*i-1), {x, 0, i}]; Sow[sc/(1-d*i)]]][[2, 1]] (* Jean-François Alcover, Mar 24 2018, after R. J. Mathar *)

Crossrefs

Primitive terms from A000108, A005789.

Sequence in context: A036077 A275765 A184975 * A319291 A265132 A080446

Adjacent sequences: A268535 A268536 A268537 * A268539 A268540 A268541

Keyword

nonn

Author

N. J. A. Sloane, Feb 24 2016

Extensions

7 more terms from R. J. Mathar, Feb 27 2018

a(17)-a(21) from Amiram Eldar, Oct 05 2025

Status

approved