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

1,4

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

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

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

Row sums give A000111(n+1).

Cf. A008282.

Sequence in context: A346520 A362925 A133611 * A019710 A118806 A328646

Adjacent sequences: A010091 A010092 A010093 * A010095 A010096 A010097

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

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

Status

approved