A350970 Triangle T(n,k) (n>=0, 0<=k<=n) read by rows: T(0,0)=T(1,1)=1; T(n,0) is the Euler number A000111(n-1) for n>=1; T(n,n-1) = T(n,n) = (n-2)! for n>=2; interior entries are given by T(n,k) = m*T(n-1,k-1)+(k+1)*T(n-1,k+1) where m = k if n+k is even or k-1 if n+k is odd.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 5, 8, 6, 6, 5, 16, 28, 40, 24, 24, 16, 61, 136, 180, 240, 120, 120, 61, 272, 662, 1232, 1320, 1680, 720, 720, 272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 1385, 7936, 24568, 56320, 83664, 129024, 100800, 120960, 40320, 40320, 7936, 50521, 176896, 408360, 814080, 1023120, 1491840, 1028160, 1209600, 362880, 362880

Offset: 0

Author

N. J. A. Sloane, Mar 03 2022

Keywords

• nonn
• tabl

Comments

Triangle connects Euler numbers on left and factorial numbers on right.

Examples

			Triangle begins:
    1,
    1,    1,
    1,    1,    1,
    1,    2,    2,    2,
    2,    5,    8,    6,     6,
    5,   16,   28,   40,    24,    24,
   16,   61,  136,  180,   240,   120,   120,
   61,  272,  662, 1232,  1320,  1680,   720,  720,
  272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040,
  ...
This may also be constructed as a square array, with entries T(n,k), n >= 1, 0 <= k, whose columns have e.g.f. equal to sec(x)+tan(x) (if k=0) and sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)) (if k>0):
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 5, 8, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, ...
5, 16, 28, 40, 24, 24, 0, 0, 0, 0, 0, 0, 0, ...
16, 61, 136, 180, 240, 120, 120, 0, 0, 0, 0, 0, 0, ...
61, 272, 662, 1232, 1320, 1680, 720, 720, 0, 0, 0, 0, 0, ...
272, 1385, 3968, 7266, 12096, 10920, 13440, 5040, 5040, 0, 0, 0, 0, ...
...
		

References

• A. Boutin, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), 252-254.
• E. Estanave, Query 2784, L'Intermédiaire des Mathématiciens, 11 (1904), pp. 117-118.

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

The initial columns are A000111, A000111, A225689, A350971.

The diagonals, reading from the right, are (essentially) A000142, A000142, A002301, A006157, A002302, A350973, A002303, A350974, A350975.

Rows sums give A156142(n-1).

Cf. A007836.

Programs

Maple

for n from 0 to 12 do
T[n]:=Array(0..n,0);
T[0,0] := 1;
T[1,0] := 1; T[1,1] := 1;
if n>1 then
  T[n,0] := T[n-1,1];
for k from 1 to n-2 do
m:=k; if ((n+k) mod 2) = 0 then m:=k-1; fi;
T[n,k] := m*T[n-1,k-1] + (k+1)*T[n-1,k+1];
od:
T[n,n-1] := (n-1)*T[n-1,n-2];
T[n,n] := T[n,n-1];
fi;
lprint( [seq(T[n,k],k=0..n)] );
od:
# second Maple program:
b:= proc(n, i) option remember; `if`(i=0,
     `if`(n=0, 1, 0), b(n, i-1)+b(n-1, n-i))
    end:
T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
     `if`(k=0, b(n-1$2), `if`(n-k<=1, (n-1)!, (k+1)*
      T(n-1, k+1)+(k-irem(1+n+k, 2))*T(n-1, k-1))))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 04 2022
# To produce the square array, N. J. A. Sloane, Mar 05 2022:
read(transforms):
myegf := (f,M) -> SERIESTOLISTMULT(series(f,x,M));
T:=proc(n,k,M) local i;
if k=0 then myegf((sec(x)+tan(x)),M)[n];
else
myegf(sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)),M)[n];
fi;
end;
[seq(T(n,0,16),n=1..5)];
for n from 1 to 8 do
lprint([seq(T(n,k,16),k=0..12)]);
od:

Mathematica

b[n_, i_] := b[n, i] = If[i == 0,
     If[n == 0, 1, 0], b[n, i - 1] + b[n - 1, n - i]];
T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
     If[k == 0, b[n - 1, n - 1], If[n - k <= 1, (n - 1)!, (k + 1)*
     T[n - 1, k + 1] + (k - Mod[1 + n + k, 2])*T[n - 1, k - 1]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2022, Alois P. Heinz *)

Formula

If we ignore the n=0 row, then the e.g.f. for column 0 is sec(x)+tan(x), and for column k >= 1 it is sec(x)*tan(x)^(k-1)*(sec(x)+tan(x)). See the initial rows of the square array in the EXAMPLES section. - N. J. A. Sloane, Mar 05 2022

abs(Sum_{k=0..n} (-1)^k * T(n,k)) = A007836(n) for n>=2. - Alois P. Heinz, Mar 04 2022