This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A243661 #43 Oct 09 2022 20:38:33
%S A243661 1,4,3,22,33,12,140,315,231,55,969,2907,3213,1547,273,7084,26565,
%T A243661 39270,28560,10200,1428,53820,242190,448500,437000,235980,66861,7752,
%U A243661 420732,2208843,4916457,6009003,4351347,1864863,437437,43263,3362260,20173560,52451256,77134200,70122000,40320150,14307150,2861430,246675
%N A243661 Triangle read by rows: the x = 1+q Narayana triangle at m=3.
%C A243661 See Novelli-Thibon (2014) for precise definition.
%H A243661 Lars Blomberg, <a href="/A243661/b243661.txt">Table of n, a(n) for n = 1..210</a> (The first 20 rows)
%H A243661 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 9.
%F A243661 From _Werner Schulte_, Nov 23 2018: (Start)
%F A243661 T(n,k) = binomial(4*n+1-k,n-k) * binomial(3*n,k-1) / n.
%F A243661 More generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(m*n,k-1) / n where m = 3.
%F A243661 Sum_{k=1..n} (-1)^k * T(n,k) = -1. (End)
%F A243661 Sum_{k = 1..n} (-1)^(k+1)*T(n,k)*binomial(x + 4*n - k + 1, 4*n - k + 1) = (x + 1) * ( Product_{k = 2..n} (x + k)^2 ) * ( Product_{k = 1..2*n+1} (x + n + k) ) / (n!*(3*n + 1)!) for n >= 1. Cf. A126216 and A243660. - _Peter Bala_, Oct 08 2022
%e A243661 Triangle begins:
%e A243661 1;
%e A243661 4, 3;
%e A243661 22, 33, 12;
%e A243661 140, 315, 231, 55;
%e A243661 969, 2907, 3213, 1547, 273;
%e A243661 7084, 26565, 39270, 28560, 10200, 1428;
%e A243661 ...
%t A243661 polrecip[P_, x_] := P /. x -> 1/x // Together // Numerator;
%t A243661 P[n_, m_] := Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}]/(m n + 1);
%t A243661 T[m_] := Reap[For[i=1, i <= 20, i++, z = polrecip[P[i, m], x] /. x -> 1+q; Sow[CoefficientList[z, q]]]][[2, 1]];
%t A243661 T[3] // Flatten (* _Jean-François Alcover_, Oct 08 2018, from PARI *)
%o A243661 (PARI)
%o A243661 N(n,m)=sum(k=0,n,binomial(m*n+1,k)*binomial((m+1)*n-k,n-k)*(1-x)^k*x^(n-k))/(m*n+1);
%o A243661 T(m)=for(i=1,20,z=subst(polrecip(N(i,m)),x,1+q);print(Vecrev(z)));
%o A243661 T(3) /* _Lars Blomberg_, Jul 17 2017 */
%Y A243661 The left column is A002293, the main diagonal is A001764.
%Y A243661 The case m=1 is A126216 or A033282 (its mirror image).
%Y A243661 The case m=2 is A243660.
%K A243661 nonn,tabl
%O A243661 1,2
%A A243661 _N. J. A. Sloane_, Jun 13 2014
%E A243661 a(22)-a(39) from _Lars Blomberg_, Jul 12 2017