1,2

Row sums are the factorial numbers (A000142). T(n,n) = A075866(n). In the Maple program define n (<=10) to obtain row n.

Alois P. Heinz, Rows n = 1..14, flattened

T(4,3) = 6 because we have 1423, 1342, 3124, 4312, 2314 and 3421.

Triangle starts:

1;

2, 0;

1, 5, 0;

3, 11, 6, 4;

1, 28, 55, 32, 4;

3, 69, 210, 330, 108, 0;

...

n:=7: with(combinat): P:=permute(n): for j from 1 to n! do c[j]:=0 od: for j from 1 to n! do if nops({seq(abs(P[j][i]-i), i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(abs(P[j][i]-i), i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i], i=1..n); # yields row n for the specified n (n<=10)

# Alternative:

b:= proc(p, s) option remember; `if`(p={}, x^nops(s),

add(b(p minus {t}, s union {abs(t-nops(p))}), t=p))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b({$1..n}, {})):

seq(T(n), n=1..9); # Alois P. Heinz, Feb 21 2019

b[p_, s_] := b[p, s] = If[p == {}, x^Length[s], Sum[b[p ~Complement~ {t}, s ~Union~ {Abs[t - Length[p]]}], {t, p}]];

T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n}]][b[Range[n], {}]];

Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Cf. A000142, A075866, A125182.

Sequence in context: A394975 A325304 A134433 * A092583 A321619 A285212

Adjacent sequences: A125180 A125181 A125182 * A125184 A125185 A125186

nonn,tabl

Emeric Deutsch, Nov 24 2006

More terms from Alois P. Heinz, Feb 27 2012

approved