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

%I A268538 #23 Jan 03 2022 20:39:15
%S A268538 1,1,2,12,107,1178,14805,203885,3002973,46573347,752521980,
%T A268538 12571607865,215925120675,3796546970232,68106673339365,
%U A268538 1243210765414512,23041656826384341
%N A268538 a(n) is the n-th prime 3-dimensional Catalan number.
%C A268538 "Prime" here is used in the sense of "primitive" or "irreducible".
%H A268538 Manuel Wettstein, <a href="http://arxiv.org/abs/1602.07235">Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers</a>, arXiv:1602.07235 [cs.CG], 2016.
%F A268538 Lemma 15 of Wettstein (2016) gives a formula in terms of the 3-dimensional Catalan numbers (A005789).
%p A268538 A005789 := proc(n)
%p A268538     2*(3*n)!/(n+2)!/(n+1)!/n! ;
%p A268538 end proc:
%p A268538 maxn := 30 :
%p A268538 Cx := add(A005789(i)*x^i,i=0..maxn) ;
%p A268538 d := 3:
%p A268538 for i from 0 to maxn do
%p A268538     coeftayl(1/Cx^(d*i-1),x=0,i) ;
%p A268538     %/(1-d*i) ;
%p A268538     printf("%d,",%) ;
%p A268538 end do: # _R. J. Mathar_, Feb 27 2018
%t A268538 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_ *)
%Y A268538 Primitive terms from A000108, A005789.
%K A268538 nonn
%O A268538 0,3
%A A268538 _N. J. A. Sloane_, Feb 24 2016
%E A268538 7 more terms from _R. J. Mathar_, Feb 27 2018