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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. 19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

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

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)

EXAMPLE

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.

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 k<n then sum((-1)^i*binomial(k, 2*i+1)*E[n-2*i-1], i=0..floor((k-1)/2)) elif k=n then E[n] else 0 fi end: seq(seq(T(n, k), k=1..n), n=1..10);

# Alternatively:

T := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else

T(n, k - 1) + T(n - 1, n - k) fi end:

for n from 1 to 6 do seq(T(n, k), k=1..n) od; # Peter 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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A035002 A032578 A035659 * A296690 A074765 A029045

Adjacent sequences:  A008279 A008280 A008281 * A008283 A008284 A008285

KEYWORD

nonn,tabl,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified February 24 22:51 EST 2021. Contains 341592 sequences. (Running on oeis4.)