A008282 -id:A008282 - cp's OEIS frontend

A173253 Partial sums of A000111.

1, 2, 3, 5, 10, 26, 87, 359, 1744, 9680, 60201, 413993, 3116758, 25485014, 224845995, 2128603307, 21520115452, 231385458428, 2636265133869, 31725150246701, 402096338484226, 5353594391608322, 74702468784746223, 1090126355291598575, 16604660518848685480

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Partial sums of Euler or up/down numbers. Partial sums of expansion of sec x + tan x. Partial sums of number of alternating permutations on n letters.

			a(22) = 1 + 1 + 1 + 2 + 5 + 16 + 61 + 272 + 1385 + 7936 + 50521 + 353792 + 2702765 + 22368256 + 199360981 + 1903757312 + 19391512145 + 209865342976 + 2404879675441 + 29088885112832 + 370371188237525 + 4951498053124096 + 69348874393137901.

Alois P. Heinz, Table of n, a(n) for n = 0..485

Cf. A000111, A000364, A000182, A008280, A008281, A008282, A010094, A059720, A008970, A109449, A162170.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, a(n-1))+ b(n, 0)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 27 2017

With[{nn=30},Accumulate[CoefficientList[Series[Sec[x]+Tan[x],{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Feb 26 2012 *)

from itertools import accumulate
def A173253(n):
    if n<=1:
        return n+1
    c, blist = 2, (0,1)
    for _ in range(n-1):
        c += (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
    return c # Chai Wah Wu, Apr 16 2023

a(n) = SUM[i=0..n] A000111(i) = SUM[i=0..n] (2^i|E(i,1/2)+E(i,1)| where E(n,x) are the Euler polynomials).
G.f.: (1 + x/Q(0))/(1-x),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
G.f.: 1/(1-x) + T(0)*x/(1-x)^2, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013
a(n) ~ 2^(n+2)*n!/Pi^(n+1). - Vaclav Kotesovec, Oct 27 2016

A236934 Triangle of Poupard numbers g_n(k) read by rows, n>=1, 1<=k<=2n-1.

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 8, 4, 0, 0, 32, 64, 80, 64, 32, 0, 0, 544, 1088, 1504, 1664, 1504, 1088, 544, 0, 0, 15872, 31744, 45440, 54784, 58112, 54784, 45440, 31744, 15872, 0, 0, 707584, 1415168, 2059264, 2576384, 2911744, 3027968, 2911744, 2576384, 2059264, 1415168, 707584, 0

Offset: 1

N. J. A. Sloane, Feb 17 2014

			Triangle begins:
1,
0, 2, 0,
0, 4, 8, 4, 0,
0, 32, 64, 80, 64, 32, 0,
0, 544, 1088, 1504, 1664, 1504, 1088, 544, 0,
...

Peter Luschny, Row(n) for n = 1..25
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).
Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., vol. 10, 1989, p. 369-374.

Cf. A000182 (row sums), A008282, A125053.

T := proc(n,k) option remember; local j;
  if n = 1 then 1
elif k = 1 then 0
elif k = 2 then 2*add(T(n-1, j), j=1..2*n-3)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1)-T(n, k-2)-4*T(n-1, k-2)
  fi end:
seq(print(seq(T(n,k), k=1..2*n-1)), n=1..6); # Peter Luschny, May 11 2014

T[n_, k_] /; 1 <= k <= 2n-1 := T[n, k] = Which[n == 1, 1, k == 1, 0, k == 2, 2 Sum[T[n-1, j], {j, 1, 2n-3}], k > n, T[n, 2n-k], True, 2 T[n, k-1] - T[n, k-2] - 4 T[n-1, k-2]]; T[, ] = 0;
Table[T[n, k], {n, 1, 7}, {k, 1, 2n-1}] // Flatten (* Jean-François Alcover, Jul 08 2019, from Maple *)

4^(-n)*sum(k=1..2*n+1, binomial(2*n,k-1)*T(n+1,k)) = A000364(n), n>=0. - Peter Luschny, May 11 2014

(-1)^n*sum(k=1..2*n+1, (-1)^(k-1)*binomial(2*n,k-1)*T(n+1,k)) = A000302(n), n>=0. - Peter Luschny, May 11 2014

More terms from Peter Luschny, May 11 2014

A278982 List of Andre permutations of the first kind.

Original entry on oeis.org

1, 12, 123, 213, 1234, 1324, 2134, 2314, 3124, 12345, 12435, 13245, 13425, 14235, 21345, 21435, 23145, 23415, 24135, 31245, 31425, 32415, 34125, 41235, 41325, 123456, 123546, 124356, 124536, 125346, 132456, 132546, 134256, 134526, 135246, 142356, 142536

Offset: 1

N. J. A. Sloane, Dec 03 2016

The permutations are encoded in decimal, hence the keyword base. - Rémy Sigrist, May 13 2021

Rémy Sigrist, Table of n, a(n) for n = 1..9679
Dominique Foata and Guo-Niu Han, André Permutation Calculus; a Twin Seidel Matrix Sequence, arXiv:1601.04371 [math.CO], 2016.
Rémy Sigrist, PARI program for A278982

Cf. A000111, A008282, A278983.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, May 13 2021

A278983 List of Andre permutations of the second kind.

Original entry on oeis.org

1, 12, 123, 312, 1234, 1423, 3124, 3412, 4123, 12345, 12534, 14235, 14523, 15234, 31245, 31524, 34125, 34512, 35124, 41235, 41523, 45123, 51234, 51423, 53412, 123456, 123645, 125346, 125634, 126345, 142356, 142635, 145236, 145623, 146235, 152346, 152634

Offset: 1

N. J. A. Sloane, Dec 03 2016

The permutations are encoded in decimal, hence the keyword base. - Rémy Sigrist, May 13 2021

Rémy Sigrist, Table of n, a(n) for n = 1..9679
Dominique Foata and Guo-Niu Han, André Permutation Calculus; a Twin Seidel Matrix Sequence, arXiv:1601.04371 [math.CO], 2016.
Rémy Sigrist, PARI program for A278983

Cf. A000111, A008282, A278982.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, May 13 2021

A006213 Number of down-up permutations of n+4 starting with n+1.

Original entry on oeis.org

0, 2, 10, 46, 224, 1202, 7120, 46366, 329984, 2551202, 21306880, 191252686, 1836652544, 18793429202, 204154071040, 2346705139006, 28459289083904, 363156549211202, 4864231397785600, 68237760828425326, 1000569392347480064, 15306487540377673202

Offset: 0

N. J. A. Sloane

Entringer numbers.

			a(1) = 2 because we have 21435 and 21534.

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
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.

Cf. A000111, A008282.

Column k=4 of A010094.

f:=sec(x)+tan(x): fser:=series(f,x=0,30): E[0]:=1: for n from 1 to 25 do E[n]:=n!*coeff(fser,x^n) od: a:=n->sum((-1)^i*binomial(n,2*i+1)*E[n+2-2*i],i=0..1+floor((n+1)/2)): seq(a(n),n=0..17);
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, 3): seq(a(n), n = 0..21); # Peter Luschny, Oct 27 2017

t[n_, 0] := If[n == 0, 1, 0]; t[n_ , k_ ] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n + 3, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

From Emeric Deutsch, May 15 2004: (Start)
a(n) = Sum_{i=0..1+floor((n+1)/2)} (-1)^i * binomial(n, 2*i+1) * E[n+2-2i], where E[j] = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers.
a(n) = T(n+3, n), where T is the triangle in A008282. (End)

More terms from Jean-François Alcover, Feb 12 2016

A006215 Number of down-up permutations of n+6 starting with n+1.

Original entry on oeis.org

0, 16, 122, 800, 5296, 36976, 275792, 2204480, 18870016, 172585936, 1681843712, 17411416160, 190939611136, 2211961358896, 26999750469632, 346419349043840, 4661658528710656, 65657186909139856, 966054350401175552, 14822897275566895520

Offset: 0

N. J. A. Sloane

Entringer numbers.

			a(1)=16 because we have 2143657, 2143756, 2153647, 2153746, 2154637, 2154736, 2163547, 2163745, 2164537, 2164735, 2165734, 2173546, 2173645, 2174536, 2174635 and 2175634.

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
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.
Index entries for sequences related to tournaments

Cf. A000111, A008282.

f:=sec(x)+tan(x): fser:=series(f,x=0,30): E[0]:=1: for n from 1 to 25 do E[n]:=n!*coeff(fser,x^n) od: a:=n->sum((-1)^i*binomial(n,2*i+1)*E[n+4-2*i],i=0..floor((n-1)/2)): seq(a(n),n=0..15);
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, 5): seq(a(n), n = 0..21); # Peter Luschny, Oct 27 2017

t[n_, 0] := If[n == 0, 1, 0]; t[n_ , k_ ] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n + 5, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

From Emeric Deutsch, May 15 2004: (Start)
a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i * binomial(n, 2*i+1) * E[n + 4 - 2*i], where E[j] = A000111(j) = j! * [x^j] (sec(x) + tan(x)) are the up/down or Euler numbers.
a(n) = T(n+5, n), where T is the triangle in A008282. (End)

More terms from Jean-François Alcover, Feb 12 2016

A006216 Number of down-up permutations of n+4 starting with 4.

Original entry on oeis.org

2, 5, 14, 46, 178, 800, 4094, 23536, 150178, 1053440, 8057774, 66750976, 595380178, 5688903680, 57975175454, 627692271616, 7195247514178, 87056789995520, 1108708685037134, 14825405274259456, 207676251991176178

Offset: 0

N. J. A. Sloane

Entringer numbers.

			a(1) = 5 because we have 41325, 41523, 42314, 42513 and 43512.

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990) 16-26.
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.

Cf. A000111. Column k=3 in A008282.

f:=sec(x)+tan(x): fser:=series(f,x=0,30): E[0]:=1: for n from 1 to 25 do E[n]:=n!*coeff(fser,x^n) od: seq(3*E[n+2]-E[n],n=0..20);

e[0] = e[1] = 1; e[n_] := 2*Sum[ 4^m*Sum[ (i-(n-1)/2)^(n-1)*Binomial[ n-2*m-1, i-m]*(-1)^(n-i-1), {i, m, (n-1)/ 2}], {m, 0, (n-2)/2}]; a[0]=2; a[n_] := 3e[n+2] - e[n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 27 2012, after Emeric Deutsch *)

{a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+4, t=0; v = vector(k, i, if( i>1, t += v[k+1-i]))); v[4])}; /* Michael Somos, Feb 03 2004 */

a(n) = 3*E(n+2) - 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
E.g.f.: 6/(cos(x)*(1 - sin(x))) - tan(x) - 4*sec(x). - Sergei N. Gladkovskii, Jun 04 2015
a(n) ~ 3*n^2 * 2^(n+4) * n! / Pi^(n+3). - Vaclav Kotesovec, Jun 04 2015

A006217 Number of down-up permutations of n+5 starting with 5.

Original entry on oeis.org

5, 16, 56, 224, 1024, 5296, 30656, 196544, 1383424, 10608976, 88057856, 786632864, 7525556224, 76768604656, 831846342656, 9541952653184, 115516079079424, 1471865234248336, 19689636672045056, 275914012819601504

Offset: 0

N. J. A. Sloane

Entringer numbers.

			a(0)=5 because we have 51324, 51423, 52314, 52413 and 53412.

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
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.

Column k=4 in A008282.

Cf. A000111.

f:=sec(x)+tan(x): fser:=series(f,x=0,35): E[0]:=1: for n from 1 to 40 do E[n]:=n!*coeff(fser,x^n) od: 5, seq(4*E[n-1]-4*E[n-3],n=5..23);

{a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+5, t=0; v = vector(k, i, if( i>1, t += v[k+1-i]))); v[5])}; /* Michael Somos, Feb 03 2004 */

a(0) = 5 and a(n) = 4*E(n+3) - 4*E(n+1) for n >= 1, where E(j) = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers. - Emeric Deutsch, May 15 2004

More terms from Emeric Deutsch, May 15 2004

A006214 Number of down-up permutations of n+5 starting with n+1.

Original entry on oeis.org

0, 5, 32, 178, 1024, 6320, 42272, 306448, 2401024, 20253440, 183194912, 1769901568, 18198049024, 198465167360, 2288729963552, 27831596812288, 355961301697024, 4777174607790080, 67129052143388192, 985743987073220608, 15098811288386497024, 240833888369219993600

Offset: 0

N. J. A. Sloane

Entringer numbers.

			a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453.

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
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).

Cf. A000111, A008282.

f:=sec(x)+tan(x): fser:=series(f,x=0,30): E[0]:=1: for n from 1 to 25 do E[n]:=n!*coeff(fser,x^n) od: a:=n->sum((-1)^i*binomial(n,2*i+1)*E[n+3-2*i],i=0..floor((n-1)/2)): seq(a(n),n=0..16);
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, 4): seq(a(n), n = 0..21); # Peter Luschny, Oct 27 2017

t[n_, 0] := If[n == 0, 1, 0]; t[n_ , k_ ] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n + 4, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+3-2i], i=0..floor((n-1)/2)), where E[j]=A000111(j)=j!*[x^j](sec(x)+tan(x)) are the up/down or Euler numbers. a(n)=T(n+4, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

More terms from Jean-François Alcover, Feb 12 2016

A008283 Read across rows of Euler-Bernoulli or Entringer triangle.

Original entry on oeis.org

1, 2, 4, 5, 10, 14, 16, 32, 46, 56, 61, 122, 178, 224, 256, 272, 544, 800, 1024, 1202, 1324, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 15872, 23536, 30656, 36976, 42272, 46366, 49136, 50521, 101042, 150178, 196544, 238816, 275792, 306448, 329984, 345856

Offset: 3

N. J. A. Sloane

			This is a sub-triangle of A008282, starting in row 3 of A008282 and then proceeding as a regular triangle.
[ 3] 1
[ 4] 2,     4
[ 5] 5,     10,     14
[ 6] 16,    32,     46,     56
[ 7] 61,    122,    178,    224,    256
[ 8] 272,   544,    800,    1024,   1202,   1324
[ 9] 1385,  2770,   4094,   5296,   6320,   7120,   7664
[10] 7936,  15872,  23536,  30656,  36976,  42272,  46366,  49136
[11] 50521, 101042, 150178, 196544, 238816, 275792, 306448, 329984, 345856

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. English version.
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.

Cf. A008282.

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:
seq(seq(T(n, k-2), k = 3..n), n = 3..11); # Peter Luschny, Feb 17 2021

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     T[n, k - 1] + T[n - 1, n - k]];
Table[Table[T[n, k - 2], {k, 3, n}], {n, 3, 11}] // Flatten (* after Peter Luschny *)

cp's OEIS Frontend

A173253 Partial sums of A000111.

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A236934 Triangle of Poupard numbers g_n(k) read by rows, n>=1, 1<=k<=2n-1.

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A278982 List of Andre permutations of the first kind.

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A278983 List of Andre permutations of the second kind.

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A006213 Number of down-up permutations of n+4 starting with n+1.

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A006215 Number of down-up permutations of n+6 starting with n+1.

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A006216 Number of down-up permutations of n+4 starting with 4.

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