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 A242819 #36 Dec 01 2017 17:43:40
%S A242819 1,1,2,6,21,3,90,30,450,270,2619,2322,99,17334,20772,2214,129114,
%T A242819 195372,38394,1067661,1958337,591543,11259,9713682,20933154,8826246,
%U A242819 443718,96393726,238789782,131367258,12450834,1036348587,2900868876,1989555210,297195804,3052323
%N A242819 Number T(n,k) of permutations of [n] with exactly k occurrences of the consecutive step pattern up, down, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/3)), read by rows.
%C A242819 T(n,k) is also the number of permutations of [n] with exactly k occurrences of the consecutive step pattern up, up, down.
%C A242819 From _Vaclav Kotesovec_, Aug 26 2014: (Start)
%C A242819 Column k is asymptotic to c(k) * (3*sqrt(3)/(2*Pi))^n * n! * n^k.
%C A242819 Conjecture: c(k) = c(0) * (c(0)-1)^k / (3^k * k!).
%C A242819 Verified numerically:
%C A242819 c(0) = 1.96650951227123825842868... = (1+exp(Pi/sqrt(3)))*sqrt(3)/(2*Pi)
%C A242819 c(1) = 0.63355004986067503869384...
%C A242819 c(2) = 0.10205535828170995196503...
%C A242819 c(3) = 0.01095971939528021798...
%C A242819 c(4) = 0.000882722753946826148...
%C A242819 c(5) = 0.00005687732922585807984...
%C A242819 c(6) = 0.000003054026651631929902...
%C A242819 c(7) = 0.0000001405593242634352116...
%C A242819 c(8) = 0.00000000566049683079281633...
%C A242819 c(9) = 0.0000000002026268159682390665...
%C A242819 c(10)= 0.00000000000652802483581788974...
%C A242819 c(20)= 1.172921625090753...*10^(-28)
%C A242819 c(30)= 1.2959323...*10^(-47)
%C A242819 c(40)= 5.0751...*10^(-68)
%C A242819 (End)
%H A242819 Alois P. Heinz, <a href="/A242819/b242819.txt">Rows n = 0..120, flattened</a>
%e A242819 T(4,1) = 3: (1,4,3,2), (2,4,3,1), (3,4,2,1).
%e A242819 Triangle T(n,k) begins:
%e A242819 : 0 : 1;
%e A242819 : 1 : 1;
%e A242819 : 2 : 2;
%e A242819 : 3 : 6;
%e A242819 : 4 : 21, 3;
%e A242819 : 5 : 90, 30;
%e A242819 : 6 : 450, 270;
%e A242819 : 7 : 2619, 2322, 99;
%e A242819 : 8 : 17334, 20772, 2214;
%e A242819 : 9 : 129114, 195372, 38394;
%e A242819 : 10 : 1067661, 1958337, 591543, 11259;
%e A242819 : 11 : 9713682, 20933154, 8826246, 443718;
%p A242819 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
%p A242819 add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+
%p A242819 add(b(u+j-1, o-j, 2), j=1..o)))
%p A242819 end:
%p A242819 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A242819 seq(T(n), n=0..15);
%t A242819 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, 2], {j, 1, o}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 10 2015, after _Alois P. Heinz_ *)
%Y A242819 Columns k=0-10 give: A177479, A246246, A246247, A246248, A246249, A246250, A246251, A246252, A246253, A246254, A246255.
%Y A242819 Row sums give: A000142.
%Y A242819 Cf. A242783, A242784, A295987.
%K A242819 nonn,tabf
%O A242819 0,3
%A A242819 _Alois P. Heinz_, May 23 2014