OFFSET
0,3
COMMENTS
T(n,k) is also the number of permutations of [n] with exactly k occurrences of the consecutive step pattern up, up, down.
From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k is asymptotic to c(k) * (3*sqrt(3)/(2*Pi))^n * n! * n^k.
Conjecture: c(k) = c(0) * (c(0)-1)^k / (3^k * k!).
Verified numerically:
c(0) = 1.96650951227123825842868... = (1+exp(Pi/sqrt(3)))*sqrt(3)/(2*Pi)
c(1) = 0.63355004986067503869384...
c(2) = 0.10205535828170995196503...
c(3) = 0.01095971939528021798...
c(4) = 0.000882722753946826148...
c(5) = 0.00005687732922585807984...
c(6) = 0.000003054026651631929902...
c(7) = 0.0000001405593242634352116...
c(8) = 0.00000000566049683079281633...
c(9) = 0.0000000002026268159682390665...
c(10)= 0.00000000000652802483581788974...
c(20)= 1.172921625090753...*10^(-28)
c(30)= 1.2959323...*10^(-47)
c(40)= 5.0751...*10^(-68)
(End)
LINKS
Alois P. Heinz, Rows n = 0..120, flattened
EXAMPLE
T(4,1) = 3: (1,4,3,2), (2,4,3,1), (3,4,2,1).
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 6;
: 4 : 21, 3;
: 5 : 90, 30;
: 6 : 450, 270;
: 7 : 2619, 2322, 99;
: 8 : 17334, 20772, 2214;
: 9 : 129114, 195372, 38394;
: 10 : 1067661, 1958337, 591543, 11259;
: 11 : 9713682, 20933154, 8826246, 443718;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+
add(b(u+j-1, o-j, 2), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 23 2014
STATUS
approved