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

1,5

COMMENTS

Triangle 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.

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, 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, Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.

Neil J. Y. Fan, 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, 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 on transform, J. Combin. Theory, 17A 44-54 1996 (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

T(n, k)=sum((-1)^i*binomial(k, 2i+1)*E[n-2i-1], i=0..floor((k-1)/2))= sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (k<n), T(n, n)=E[n]. T(n, n)=E[n]; T(n, k)=sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (k<n), T(n, n)=E[n]. where E(j)=A000111(j)=j!*[x^j]((sec(x)+tan(x)) are the up/down or Euler numbers. - Emeric Deutsch, May 15 2004

EXAMPLE

T(4,3)=5 because we have 41325,41523,42314,42513 and 43512.

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

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

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 16 00:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)