A010094 Triangle of Euler-Bernoulli or Entringer numbers.

1, 1, 1, 2, 2, 1, 5, 5, 4, 2, 16, 16, 14, 10, 5, 61, 61, 56, 46, 32, 16, 272, 272, 256, 224, 178, 122, 61, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385, 50521, 50521, 49136, 46366, 42272, 36976, 30656, 23536, 15872, 7936, 353792

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Comments

T(n, k) is the number of up-down permutations of n starting with k where 1 <= k <= n. - Michael Somos, Jan 20 2020

Examples

			From _Vincenzo Librandi_, Aug 13 2013: (Start)
Triangle begins:
     1;
     1,    1;
     2,    2,    1;
     5,    5,    4,    2;
    16,   16,   14,   10,    5;
    61,   61,   56,   46,   32,   16;
   272,  272,  256,  224,  178,  122,   61;
  1385, 1385, 1324, 1202, 1024,  800,  544,  272;
  7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385;
  ... (End)
Up-down permutations for n = 4 are k = 1: 1324, 1423; k = 2: 2314, 2413; k = 3: 3411; k = 4: none. - _Michael Somos_, Jan 20 2020
		

References

• R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.

Links

• Alois P. Heinz, Rows n = 1..150, flattened (first 51 rows from Vincenzo Librandi)
• B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
• D. Foata and G.-N. Han, Secant Tree Calculus, arXiv preprint arXiv:1304.2485 [math.CO], 2013.
• Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
• Foata, Dominique; Han, Guo-Niu; Strehl, Volker The Entringer-Poupard matrix sequence. Linear Algebra Appl. 512, 71-96 (2017). example 4.4
• M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
• C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

Crossrefs

Columns k=1,3-4 give: A000111, A006212, A006213.

Row sums give A000111(n+1).

Cf. A008282.

Programs

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
T:= (n, k)-> b(n-k+1, k-1):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jun 03 2020

Mathematica

e[0, 0] = 1; e[, 0] = 0; e[n, k_] := e[n, k] = e[n, k-1] + e[n-1, n-k]; Join[{1}, Table[e[n, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten] (* Jean-François Alcover, Aug 13 2013 *)

PARI

{T(n, k) = if( n < 1 || k >= n, k == 1 && n == 1, T(n, k+1) + T(n-1, n-k))}; /* Michael Somos, Jan 20 2020 */

Formula

T(1, 1) = 1; T(n, n) = 0 if n > 1; T(n, k) = T(n, k+1) + T(n-1, n-k) if 1 <= k < n. - Michael Somos, Jan 20 2020

Extensions

More terms from Will Root (crosswind(AT)bright.net), Oct 08 2001

Irregular zeroth row deleted by N. J. A. Sloane, Jun 04 2020