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

0,8

Comments

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

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

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

Example

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, ...
...

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 *)

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.

Sequence in context: A195601 A352723 A222255 * A126788 A098789 A162489

Adjacent sequences: A350967 A350968 A350969 * A350971 A350972 A350973

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 03 2022

Status

approved