A099959 -id:A099959 - cp's OEIS frontend

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

N. J. A. Sloane

			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.

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

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

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

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

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

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 *)

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

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

A099961 Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every third row repeating the final term.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 5, 10, 13, 13, 23, 28, 28, 51, 64, 64, 64, 128, 179, 207, 207, 386, 514, 578, 578, 1092, 1478, 1685, 1685, 1685, 3370, 4848, 5940, 6518, 6518, 12458, 17306, 20676, 22361, 22361, 43037, 60343, 72801, 79319, 79319, 79319

Offset: 0

N. J. A. Sloane, Nov 13 2004

...

			Triangle begins
   1;
   1,
   1,  1;
   1,  2,
   2,  3,
   3,  5,  5;
   5, 10, 13,
  13, 23, 28,
  28, 51, 64, 64;

Reinhard Zumkeller, Rows n=0..120 of triangle, flattened

First column (and row sums) gives A099962. Cf. A099963, A099967.

If an extra term is added to /every/ row we get A008282. Cf. A099959.

a099961 n k = a099961_tabl !! n !! k
a099961_row n = a099961_tabl !! n
a099961_tabl = map snd $ iterate f (0,[1]) where
   f (s,xs) = (s+1, if s `mod` 3 == 1 then zs ++ [last zs] else zs)
     where zs = scanl1 (+) (reverse xs)
-- Reinhard Zumkeller, Dec 28 2011

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n,p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j],j=1..i): vector(n,q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j],j=1..i) else sum(a[j],j=1..n) fi end: vector(n+1,q) end: R[0]:=vector(1,1): for n from 1 to 19 do if n mod 3 = 0 or n mod 3 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 19 do evalm(R[n]) od; # program yields the successive rows # Emeric Deutsch, Nov 16 2004

r[0] = {1}; r[n_] := r[n] = Join[ a = Accumulate[ Reverse[r[n-1]]], If[Mod[n, 3] == 2, {Last[a]}, {}]]; Flatten[ Table[r[n], {n, 0, 15}]](* Jean-François Alcover, Mar 13 2012 *)

More terms from Emeric Deutsch, Nov 16 2004

A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 8, 17, 56, 155, 608, 2073, 9440, 38227, 198272, 929569, 5410688, 28820619, 186043904, 1109652905, 7867739648, 51943281731, 401293838336, 2905151042481, 24290513745920, 191329672483963, 1721379917619200, 14655626154768697, 141174819474169856

Offset: 0

N. J. A. Sloane, Nov 13 2004

First column (also row sums) of triangle in A099959.

Number of ascent sequences of length n without level steps and with alternating ascents and descents. a(6) = 8: 010101, 010102, 010103, 010201, 010202, 010203, 010212, 010213. - Alois P. Heinz, Oct 27 2017

Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 220, answer to exercise 174, Addison-Wesley, 2009.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Catalin Zara, Cardinality of l_1-Segments and Genocchi Numbers, arXiv:1304.5798 [math.CO] (2013)

Cf. A022493, A099959, A001469, A005439, A138265, A294281.

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n,p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j],j=1..i): vector(n,q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j],j=1..i) else sum(a[j],j=1..n) fi end: vector(n+1,q) end: R[0]:=vector(1,1): for n from 1 to 30 do if n mod 2 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: seq(R[n][1],n=0..30); # Emeric Deutsch

g1 = Table[2*(4^n-1)*BernoulliB[2*n] // Abs, {n, 0, 13}]; g2 = Table[2*(-1)^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}], {n, 0, 13}]; Riffle[g1, g2] // Rest (* Jean-François Alcover, May 23 2013 *)

# Algorithm of L. Seidel (1877)
def A099960_list(n) :
    D = [0]*(n//2+3); D[1] = 1
    R = []; b = True; h = 1
    for i in (1..n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k+1]
            R.append(D[1]); h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1])
        b = not b
    return R
A099960_list(27)  # Peter Luschny, Apr 30 2012

a(n) ~ 2^(5/2) * n^(n+3/2) / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Sep 10 2014

More terms from Emeric Deutsch, Nov 16 2004

A099964 Triangle read by rows: The n-th row is constructed by forming the partial sums of the previous row, reading from the right and if n is a triangular number repeating the final term.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 3, 6, 8, 8, 14, 17, 17, 31, 39, 39, 39, 78, 109, 126, 126, 235, 313, 352, 352, 665, 900, 1026, 1026, 1926, 2591, 2943, 2943, 2943, 5886, 8477, 10403, 11429, 11429, 21832, 30309, 36195, 39138, 39138, 75333, 105642, 127474, 138903

Offset: 0

N. J. A. Sloane, Nov 13 2004

...

			Triangle begins
     1;
     1,    1;
     1,    2,
     2,    3,    3;
     3,    6,    8,
     8,   14,   17,
    17,   31,   39,   39;
    39,   78,  109,  126,
   126,  235,  313,  352,
   352,  665,  900, 1026,
  1026, 1926, 2591, 2943, 2943;

Reinhard Zumkeller, Rows n = 0..500 of triangle, flattened

First column (and row sums) gives A099965. Cf. A099966, A099968.
If an extra term is added to /every/ row we get A008282. Cf. A099959, A099961.
Cf. A010054.

a099964 n k = a099964_tabf !! n !! k
a099964_row n = a099964_tabf !! n
a099964_tabf = scanl f [1] $ tail a010054_list where
   f row t = if t == 1 then row' ++ [last row'] else row'
           where row' = scanl1 (+) $ reverse row
-- Reinhard Zumkeller, May 02 2012

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n,p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j],j=1..i): vector(n,q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j],j=1..i) else sum(a[j],j=1..n) fi end: vector(n+1,q) end: tr:={seq(n*(n+1)/2,n=1..30)}: R[0]:=vector(1,1): for n from 1 to 15 do if member(n,tr)=false then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: for n from 0 to 15 do evalm(R[n]) od; # Emeric Deutsch, Nov 16 2004

triQ[n_] := Reduce[ n == k(k+1)/2, k, Integers] =!= False; row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = (ro = Accumulate[ Reverse[ row[n-1]]]; If[triQ[n], Append[ ro, Last[ro] ], ro]); Flatten[ Table[ row[n], {n, 0, 13}]](* Jean-François Alcover, Nov 24 2011 *)

More terms from Emeric Deutsch, Nov 16 2004

A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 8, 14, 17, 17, 56, 104, 138, 155, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569, 5410688, 10623104, 15448416, 19716064, 23281432, 26031912

Offset: 0

Peter Luschny, Jan 03 2018

			The triangle starts:
0: [     1]
1: [     1,      1]
2: [     2,      3,      3]
3: [     8,     14,     17,     17]
4: [    56,    104,    138,    155,    155]
5: [   608,   1160,   1608,   1918,   2073,   2073]
6: [  9440,  18272,  25944,  32008,  36154,  38227,  38227]
7: [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569]

Row sums are A005439 with offset 0.
T(n,0) = A005439 with A005439(0) = 1.
T(n,n) = A110501 with offset 0.
Cf. A001469, A014781, A099959, A226158.

function A297703Triangle(len::Int)
    A = fill(BigInt(0), len+2); A[2] = 1
    for n in 2:len+1
        for k in n:-1:2 A[k] += A[k+1] end
        for k in 2: 1:n A[k] += A[k-1] end
        println(A[2:n])
    end
end
println(A297703Triangle(9))

from functools import cache
@cache
def T(n):  # returns row n
    if n == 0: return [1]
    row = [0] + T(n - 1) + [0]
    for k in range(n, 0, -1): row[k] += row[k + 1]
    for k in range(2, n + 2): row[k] += row[k - 1]
    return row[1:]
for n in range(9): print(T(n))  # Peter Luschny, Jun 03 2022

cp's OEIS Frontend

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.

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A099961 Triangle read by rows: Each row is constructed by forming the partial sums of the previous row, reading from the right and at every third row repeating the final term.

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A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439.

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A099964 Triangle read by rows: The n-th row is constructed by forming the partial sums of the previous row, reading from the right and if n is a triangular number repeating the final term.

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Haskell

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A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

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