A008282 Triangle of Euler-Bernoulli or Entringer numbers read by rows: T(n,k) is the number of down-up permutations of n+1 starting with k+1.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy
• nice

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins
   1
   1  1
   1  2  2
   2  4  5  5
   5 10 14 16 16
  16 32 46 56 61 61
  ...
Each row is constructed by forming the partial sums of the previous row, reading from the right and repeating the final term.
T(4,3) = 5 because we have 41325, 41523, 42314, 42513 and 43512. All these permutations have length n+1 = 5, start with k+1 = 4, and they are down-up permutations.
		

References

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

Links

• Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
• V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
• J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
• B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
• Carolina Benedetti, Rafael S. González D’León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, Alejandro H. Morales, and Martha Yip, The volume of the caracol polytope, Séminaire Lotharingien de Combinatoire XX (2018), Article #YY, Proceedings of the 30th Conference on Formal Power, Series and Algebraic Combinatorics (Hanover), 2018.
• Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
• Neil J. Y. Fan and Liao He, The Complete cd-Index of Boolean Lattices, Electron. J. Combin., 22 (2015), #P2.45.
• Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.
• Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, European Journal of Combinatorics, 42 (2014), 243-260. [See Corollary 1.3. In Eq. (1.10), the power of x should be k-1 rather than k.]
• Dominique Foata and Guo-Niu Han, André Permutation Calculus; a Twin Seidel Matrix Sequence, arXiv:1601.04371 [math.CO], 2016.
• B. Gourevitch, L'univers de Pi.
• G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
• Peter Luschny, An old operation on sequences: the Seidel transform.
• 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.
• C. Poupard, Two other interpretations of the Entringer numbers, Eur. J. Combinat. 18 (1997) 939-943.
• Wikipedia, Boustrophedon transform.
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A010094, A000111, A099959, A009766, A236935.

Programs

Haskell

a008282 n k = a008282_tabl !! (n-1) !! (k-1)
a008282_row n = a008282_tabl !! (n-1)
a008282_tabl = iterate f [1] where
   f xs = zs ++ [last zs] where zs = scanl1 (+) (reverse xs)
-- Reinhard Zumkeller, Dec 28 2011

Maple

f:=series(sec(x)+tan(x),x=0,25): E[0]:=1: for n from 1 to 20 do E[n]:=n!*coeff(f,x^n) od: T:=proc(n,k) if kPeter Luschny, Aug 03 2017
# Third program:
T := proc(n, k) local w: if 0 = n mod 2 then w := coeftayl(cos(x)/cos(x + y), [x, y] = [0, 0], [n - k, k]): end if: if 1 = n mod 2 then w := coeftayl(sin(x)/cos(x + y), [x, y] = [0, 0], [k, n - k]): end if: w*(n - k)!*k!: end proc:
for n from 1 to 6 do seq(T(n,k), k=1..n) od; # Petros Hadjicostas, Feb 17 2021

Mathematica

ro[1] = {1}; ro[n_] := ro[n] = (s = Accumulate[ Reverse[ ro[n-1]]]; Append[ s, Last[s]]); Flatten[ Table[ ro[n], {n, 1, 10}]] (* Jean-François Alcover, Oct 03 2011 *)
nxt[lst_]:=Module[{lst2=Accumulate[Reverse[lst]]},Flatten[Join[ {lst2,Last[ lst2]}]]]; Flatten[NestList[nxt,{1},10]] (* Harvey P. Dale, Aug 17 2014 *)

Formula

From Emeric Deutsch, May 15 2004: (Start)

Let E[j] = A000111(j) = j! * [x^j](sec(x) + tan(x)) be the up/down or Euler numbers. For 1 <= k < n,

T(n, k) = Sum_{i=0..floor((k-1)/2)} (-1)^i * binomial(k, 2*i+1) * E[n-2*i-1];

T(n,k) = Sum_{i=0..floor((n-k)/2)} (-1)^i * binomial(n-k, 2*i) * E[n-2*i];

T(n, k) = Sum_{i=0..floor((n-k)/2)} (-1)^i * binomial(n-k, 2*i) * E[n-2*i]; and

T(n, n) = E[n] for n >= 1. (End)

From Petros Hadjicostas, Feb 17 2021: (Start)

If n is even, then T(n,k) = k!*(n-k)!*[x^(n-k),y^k] cos(x)/cos(x + y).

If n is odd, then T(n,k) = k!*(n-k)!*[x^k,y^(n-k)] sin(x)/cos(x + y).

(These were adapted and corrected from the formulas in Corollary 1.3 in Foata and Guo-Niu Han (2014).) (End)

Comment from Masanobu Kaneko: (Start)

A generating function that applies for all n, both even and odd:

Sum_{n=0..oo} Sum_{k=0..n} T(n,k) x^(n-k)/(n-k)! * y^k/k! = {cos x + sin y}/cos(x + y).

(End) - N. J. A. Sloane, Feb 06 2022

Extensions

Example and Formula sections edited by Petros Hadjicostas, Feb 17 2021