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

Offset: 0

Author

N. J. A. Sloane, Feb 24 2016

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Primitive terms from A000108, A005789.

Programs

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 *)

Formula

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

Extensions

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