OFFSET
1,4
COMMENTS
LINKS
Alois P. Heinz, Rows n = 1..150, flattened
E. Deutsch and W. P. Johnson, Create your own permutation statistics, Math. Mag., 77, 130-134, 2004.
FORMULA
T(n, k) = n!*[(k+1)*E(k)-E(k+1)]/(k+1)! for k<n and T(n, n) = E(n), where tan(x)+sec(x) = Sum_{n>=0} [E(n)x^n/n!] (i.e., E(n) = A000111(n)).
Sum_{k=0..n} (k+1) * T(n,k) = A230960(n). - Alois P. Heinz, Apr 27 2023
EXAMPLE
T(4,3)=3 because 1432, 2431, 3421 are the only permutations of [4] in which exactly the first 3 entries satisfy the up-down property.
Triangle starts:
1;
1, 1;
3, 1, 2;
12, 4, 3, 5;
60, 20, 15, 9, 16;
360, 120, 90, 54, 35, 61;
...
MAPLE
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
end:
E:= n-> b(n, 0):
T:= (n, k)-> `if`(n=k, E(n), n!*((k+1)*E(k)-E(k+1))/(k+1)!):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Aug 12 2016
MATHEMATICA
b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[o - 1 + j, u - j], {j, 1, u}]]; e[n_] := b[n, 0]; T[n_, k_] := If[n == k, e[n], n!*((k + 1)*e[k] - e[k + 1])/(k + 1)!]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch and Warren P. Johnson (wjohnson(AT)bates.edu), Apr 10 2004
STATUS
approved