A109449 -id:A109449 - cp's OEIS frontend

A343845 a(n) = Sum_{k=0..floor(n/2)} A109449(n-k, k).

1, 1, 2, 4, 9, 27, 93, 392, 1898, 10493, 64885, 443916, 3326317, 27085015, 238073306, 2246348560, 22643042325, 242808804441, 2759740869777, 33138397797908, 419171443909394, 5570771017483187, 77603014042711369, 1130712331125929112, 17198408830271090233

Offset: 0

Peter Luschny, May 06 2021

nonn

Cf. A109449.

seq(add(A109449(n-k, k), k = 0..n/2), n = 0..25);

Table[Sum[Binomial[n-k,k] * 2^(n-2*k) * Abs[EulerE[n-2*k,1/2] + EulerE[n-2*k,1]], {k,0,Floor[n/2]}] - (1 + (-1)^n)/2, {n,0,25}] (* Vaclav Kotesovec, May 06 2021 *)

a(n) ~ 2^(n+2) * n! / Pi^(n+1). - Vaclav Kotesovec, May 06 2021

A000111 Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, 370371188237525, 4951498053124096, 69348874393137901, 1015423886506852352, 15514534163557086905, 246921480190207983616, 4087072509293123892361

Offset: 0

N. J. A. Sloane

Number of linear extensions of the "zig-zag" poset. See ch. 3, prob. 23 of Stanley. - Mitch Harris, Dec 27 2005
Number of increasing 0-1-2 trees on n vertices. - David Callan, Dec 22 2006
Also the number of refinements of partitions. - Heinz-Richard Halder (halder.bichl(AT)t-online.de), Mar 07 2008
The ratio a(n)/n! is also the probability that n numbers x1,x2,...,xn randomly chosen uniformly and independently in [0,1] satisfy x1 > x2 < x3 > x4 < ... xn. - Pietro Majer, Jul 13 2009
For n >= 2, a(n-2) = number of permutations w of an ordered n-set {x_1 < ... x_n} with the following properties: w(1) = x_n, w(n) = x_{n-1}, w(2) > w(n-1), and neither any subword of w, nor its reversal, has the first three properties. The count is unchanged if the third condition is replaced with w(2) < w(n-1). - Jeremy L. Martin, Mar 26 2010
A partition of zigzag permutations of order n+1 by the smallest or the largest, whichever is behind. This partition has the same recurrent relation as increasing 1-2 trees of order n, by induction the bijection follows. - Wenjin Woan, May 06 2011
As can be seen from the asymptotics given in the FORMULA section, one has lim_{n->oo} 2*n*a(n-1)/a(n) = Pi; see A132049/A132050 for the simplified fractions. - M. F. Hasler, Apr 03 2013
a(n+1) is the sum of row n in triangle A008280. - Reinhard Zumkeller, Nov 05 2013
M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon (2011) give a far-reaching generalization of the bijection between Euler numbers and alternating permutations. - N. J. A. Sloane, Jul 09 2015
Number of treeshelves avoiding pattern T321. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link, see A278678 for more definitions and examples. - Sergey Kirgizov, Dec 24 2016
Number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that no three terms are equal. [Theorem 7 of Corteel, Martinez, Savage, and Weselcouch] - Eric M. Schmidt, Jul 17 2017
Number of self-dual edge-labeled trees with n vertices under "mind-body" duality.  Also number of self-dual rooted edge-labeled trees with n vertices.  See my paper linked below. - Nikos Apostolakis, Aug 01 2018
The ratio a(n)/n! is the volume of the convex polyhedron defined as the set of (x_1,...,x_n) in [0,1]^n such that x_i + x_{i+1} <= 1 for every 1 <= i <= n-1; see the solutions by Macdonald and Nelsen to the Amer. Math. Monthly problem referenced below. - Sanjay Ramassamy, Nov 02 2018
Number of total cyclic orders on {0,1,...,n} such that the triple (i-1,i,i+1) is positively oriented for every 1 <= i <= n-1; see my paper on cyclic orders linked below. - Sanjay Ramassamy, Nov 02 2018
The number of binary, rooted, unlabeled histories with n+1 leaves (following the definition of Rosenberg 2006). Also termed Tajima trees, Tajima genealogies, or binary, rooted, unlabeled ranked trees (Palacios et al. 2015). See Disanto & Wiehe (2013) for a proof. - Noah A Rosenberg, Mar 10 2019
From Gus Wiseman, Dec 31 2019: (Start)
Also the number of non-isomorphic balanced reduced multisystems with n + 1 distinct atoms and maximum depth. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The labeled version is A006472. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 5 multisystems are (commas elided):
  {1}  {12}  {{1}{23}}  {{{1}}{{2}{34}}}  {{{{1}}}{{{2}}{{3}{45}}}}
                        {{{12}}{{3}{4}}}  {{{{1}}}{{{23}}{{4}{5}}}}
                                          {{{{1}{2}}}{{{3}}{{45}}}}
                                          {{{{1}{23}}}{{{4}}{{5}}}}
                                          {{{{12}}}{{{3}}{{4}{5}}}}
Also the number of balanced reduced multisystems with n + 1 equal atoms and maximum depth. This is possibly the meaning of Heinz-Richard Halder's comment (see also A002846, A213427, A265947). The non-maximum-depth version is A318813. For example, the a(0) = 1 through a(4) = 5 multisystems are (commas elided):
  {1}  {11}  {{1}{11}}  {{{1}}{{1}{11}}}  {{{{1}}}{{{1}}{{1}{11}}}}
                        {{{11}}{{1}{1}}}  {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}
(End)
With s_n denoting the sum of n independent uniformly random numbers chosen from [-1/2,1/2], the probability that the closest integer to s_n is even is exactly 1/2 + a(n)/(2*n!). (See Hambardzumyan et al. 2023, Appendix B.) - Suhail Sherif, Mar 31 2024
The number of permutations of size n+1 that require exactly n passes through a stack (i.e. have reverse-tier n-1) with an algorithm that prioritizes outputting the maximum possible prefix of the identity in a given pass and reverses the remainder of the permutation for prior to the next pass. - Rebecca Smith, Jun 05 2024

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 61*x^6 + 272*x^7 + 1385*x^8 + ...
Sequence starts 1,1,2,5,16,... since possibilities are {}, {A}, {AB}, {ACB, BCA}, {ACBD, ADBC, BCAD, BDAC, CDAB}, {ACBED, ADBEC, ADCEB, AEBDC, AECDB, BCAED, BDAEC, BDCEA, BEADC, BECDA, CDAEB, CDBEA, CEADB, CEBDA, DEACB, DEBCA}, etc. - _Henry Bottomley_, Jan 17 2001

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Index entries for "core" sequences
Index entries for sequences related to boustrophedon transform

Cf. A000364 (secant numbers), A000182 (tangent numbers).
Related triangles: A008280, A008281, A008282, A008970, A010094, A059720, A177762.
Cf. A181937 for n-alternating permutations.
Cf. A109449 for an extension to an exponential Riordan array.
Column k=1 of A010094, A229892, A258829, A262124, A275784.
Column k=2 of A250261.
Cf. A002105, A186365.
For 0-1-2 trees with n nodes and k leaves, see A301344.
Matula-Goebel numbers of 0-1-2 trees are A292050.
An overview over generalized Euler numbers gives A349264.
Cf. A000258, A000311, A002846, A005121, A318813, A320270, A330474, A330665, A330679, A340315.

a000111 0 = 1
a000111 n = sum $ a008280_row (n - 1)
-- Reinhard Zumkeller, Nov 01 2013

A000111 := n-> n!*coeff(series(sec(x)+tan(x),x,n+1), x, n);
s := series(sec(x)+tan(x), x, 100): A000111 := n-> n!*coeff(s, x, n);
A000111:=n->piecewise(n mod 2=1,(-1)^((n-1)/2)*2^(n+1)*(2^(n+1)-1)*bernoulli(n+1)/(n+1),(-1)^(n/2)*euler(n)):seq(A000111(n),n=0..30); A000111:=proc(n) local k: k:=floor((n+1)/2): if n mod 2=1 then RETURN((-1)^(k-1)*2^(2*k)*(2^(2*k)-1)*bernoulli(2*k)/(2*k)) else RETURN((-1)^k*euler(2*k)) fi: end:seq(A000111(n),n=0..30); (C. Ronaldo)
T := n -> 2^n*abs(euler(n,1/2)+euler(n,1)): # Peter Luschny, Jan 25 2009
S := proc(n,k) option remember; if k=0 then RETURN(`if`(n=0,1,0)) fi; S(n,k-1)+S(n-1,n-k) end:
A000364 := n -> S(2*n,2*n);
A000182 := n -> S(2*n+1,2*n+1);
A000111 := n -> S(n,n); # Peter Luschny, Jul 29 2009
a := n -> 2^(n+2)*n!*(sum(1/(4*k+1)^(n+1), k = -infinity..infinity))/Pi^(n+1):
1, seq(a(n), n = 1..22); # Emeric Deutsch, Aug 17 2009
# alternative Maple program:
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, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 29 2015

n=22; CoefficientList[Series[(1+Sin[x])/Cos[x], {x, 0, n}], x] * Table[k!, {k, 0, n}] (* Jean-François Alcover, May 18 2011, after Michael Somos *)
a[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n+1)*(2^(n+1)-1)*BernoulliB[n+1])/(n+1)]]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 09 2012, after C. Ronaldo *)
ee = Table[ 2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, 26}]; Table[ Differences[ee, n] // First // Abs, {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013, after Paul Curtz *)
a[ n_] := If[ n < 0, 0, (2 I)^n If[ EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]]; (* Michael Somos, Aug 15 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], With[{m = n - 1}, m! SeriesCoefficient[ 1 / (1 - Sin[x]), {x, 0, m}]]]; (* Michael Somos, Aug 15 2015 *)
s[0] = 1; s[] = 0; t[n, 0] := s[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0](* Jean-François Alcover, Feb 12 2016 *)
a[n_] := If[n == 0, 1, 2*Abs[PolyLog[-n, I]]]; (* Jean-François Alcover, Dec 02 2023, after M. F. Hasler *)
a[0] := 1; a[1] := 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]; Map[a, Range[0, 26]] (* Oliver Seipel, May 24 2024 after Peter Bala *)
a[0] := 1; a[1] := 1; a[n_] := a[n] = 1/(n (n-1)) Sum[a[n-1-k] a[k] k, {k, 1, n-1}]; Map[#! a[#]&, Range[0, 26]] (* Oliver Seipel, May 27 2024 *)

a(n):=sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*stirling2(n,j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n) else 0),k,1,n); /* Vladimir Kruchinin, Aug 19 2010 */

a(n):=if n<2 then 1 else 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); /* Vladimir Kruchinin, Aug 09 2011 */

{a(n) = if( n<1, n==0, n--; n! * polcoeff( 1 / (1 - sin(x + x * O(x^n))), n))}; \\ Michael Somos, Feb 03 2004

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

{a(n) = local(an); if( n<1, n>=0, an = vector(n+1, m, 1); for( m=2, n, an[m+1] = sum( k=0, m-1, binomial(m-1, k) * an[k+1] * an[m-k]) / 2); an[n+1])}; \\ Michael Somos, Feb 03 2004

z='z+O('z^66); egf = (1+sin(z))/cos(z); Vec(serlaplace(egf)) \\ Joerg Arndt, Apr 30 2011

A000111(n)={my(k);sum(m=0,n\2,(-1)^m*sum(j=0,k=n+1-2*m,binomial(k,j)*(-1)^j*(k-2*j)^(n+1))/k>>k)}  \\ M. F. Hasler, May 19 2012

A000111(n)=if(n,2*abs(polylog(-n,I)),1)  \\ M. F. Hasler, May 20 2012

# requires python 3.2 or higher
from itertools import accumulate
A000111_list, blist = [1,1], [1]
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    A000111_list.append(sum(blist)) # Chai Wah Wu, Jan 29 2015

from mpmath import *
mp.dps = 150
l = chop(taylor(lambda x: sec(x) + tan(x), 0, 26))
[int(fac(i) * li) for i, li in enumerate(l)]  # Indranil Ghosh, Jul 06 2017

from sympy import bernoulli, euler
def A000111(n): return abs(((1<Chai Wah Wu, Nov 13 2024

# Algorithm of L. Seidel (1877)
def A000111_list(n) :
    R = []; A = {-1:0, 0:1}; k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        R.append(Am)
    return R
A000111_list(22) # Peter Luschny, Mar 31 2012 (revised Apr 24 2016)

E.g.f.: (1+sin(x))/cos(x) = tan(x) + sec(x).
E.g.f. for a(n+1) is 1/(cos(x/2) - sin(x/2))^2 = 1/(1-sin(x)) = d/dx(sec(x) + tan(x)).
E.g.f. A(x) = -log(1-sin(x)), for a(n+1). - Vladimir Kruchinin, Aug 09 2010
O.g.f.: A(x) = 1+x/(1-x-x^2/(1-2*x-3*x^2/(1-3*x-6*x^2/(1-4*x-10*x^2/(1-... -n*x-(n*(n+1)/2)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
E.g.f. A(x) = y satisfies 2y' = 1 + y^2. - Michael Somos, Feb 03 2004
a(n) = P_n(0) + Q_n(0) (see A155100 and A104035), defining Q_{-1} = 0. Cf. A156142.
2*a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k).
Asymptotics: a(n) ~ 2^(n+2)*n!/Pi^(n+1). For a proof, see for example Flajolet and Sedgewick.
a(n) = (n-1)*a(n-1) - Sum_{i=2..n-2} (i-1)*E(n-2, n-1-i), where E are the Entringer numbers A008281. - Jon Perry, Jun 09 2003
a(2*k) = (-1)^k euler(2k) and a(2k-1) = (-1)^(k-1)2^(2k)(2^(2k)-1) Bernoulli(2k)/(2k). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 17 2005
|a(n+1) - 2*a(n)| = A000708(n). - Philippe Deléham, Jan 13 2007
a(n) = 2^n|E(n,1/2) + E(n,1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
a(n) = 2^(n+2)*n!*S(n+1)/(Pi)^(n+1), where S(n) = Sum_{k = -inf..inf} 1/(4k+1)^n (see the Elkies reference). - Emeric Deutsch, Aug 17 2009
a(n) = i^(n+1) Sum_{k=1..n+1} Sum_{j=0..k} binomial(k,j)(-1)^j (k-2j)^(n+1) (2i)^(-k) k^{-1}. - Ross Tang (ph.tchaa(AT)gmail.com), Jul 28 2010
a(n) = sum((if evenp(n+k) then (-1)^((n+k)/2)*sum(j!*Stirling2(n,j)*2^(1-j)*(-1)^(n+j-k)*binomial(j-1,k-1),j,k,n) else 0),k,1,n), n>0. - Vladimir Kruchinin, Aug 19 2010
If n==1(mod 4) is prime, then a(n)==1(mod n); if n==3(mod 4) is prime, then a(n)==-1(mod n). - Vladimir Shevelev, Aug 31 2010
For m>=0, a(2^m)==1(mod 2^m); If p is prime, then a(2*p)==1(mod 2*p). - Vladimir Shevelev, Sep 03 2010
From Peter Bala, Jan 26 2011: (Start)
a(n) = A(n,i)/(1+i)^(n-1), where i = sqrt(-1) and {A(n,x)}n>=1 = [1,1+x,1+4*x+x^2,1+11*x+11*x^2+x^3,...] denotes the sequence of Eulerian polynomials.
Equivalently, a(n) = i^(n+1)*Sum_{k=1..n} (-1)^k*k!*Stirling2(n,k) * ((1+i)/2)^(k-1) = i^(n+1)*Sum_{k = 1..n} (-1)^k*((1+i)/2)^(k-1)* Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*j^n.
This explicit formula for a(n) can be used to obtain congruence results. For example, for odd prime p, a(p) = (-1)^((p-1)/2) (mod p), as noted by Vladimir Shevelev above.
For the corresponding type B results see A001586. For the corresponding results for plane increasing 0-1-2 trees see A080635.
For generalized Eulerian, Stirling and Bernoulli numbers associated with the zigzag numbers see A145876, A147315 and A185424, respectively. For a recursive triangle to calculate a(n) see A185414.
(End)
a(n) = I^(n+1)*2*Li_{-n}(-I) for n > 0. Li_{s}(z) is the polylogarithm. - Peter Luschny, Jul 29 2011
a(n) = 2*Sum_{m=0..(n-2)/2} 4^m*(Sum_{i=m..(n-1)/2} (i-(n-1)/2)^(n-1)*binomial(n-2*m-1,i-m)*(-1)^(n-i-1)), n > 1, a(0)=1, a(1)=1. - Vladimir Kruchinin, Aug 09 2011
a(n) = D^(n-1)(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1-x^2)*d/dx. Cf. A006154. a(n) equals the alternating sum of the nonzero elements of row n-1 of A196776. This leads to a combinatorial interpretation for a(n); for example, a(4*n+2) gives the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 1 (mod 4), minus the number of ordered set partitions of 4*n+1 into k odd-sized blocks, k = 3 (mod 4). Cf A002017. - Peter Bala, Dec 06 2011
From Sergei N. Gladkovskii, Nov 14 2011 - Dec 23 2013: (Start)
Continued fractions:
E.g.f.: tan(x) + sec(x) = 1 + x/U(0); U(k) = 4k+1-x/(2-x/(4k+3+x/(2+x/U(k+1)))).
E.g.f.: for a(n+1) is E(x) = 1/(1-sin(x)) = 1 + x/(1 - x + x^2/G(0)); G(k) = (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1).
E.g.f.: for a(n+1) is E(x) = 1/(1-sin(x)) = 1/(1 - x/(1 + x^2/G(0))) ; G(k) = 8*k+6-x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)).
E.g.f.: for a(n+1) is E(x) = 1/(1 - sin(x)) = 1/(1 - x*G(0)); G(k) = 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))).
E.g.f.: for a(n+1) is E(x) = 1/(1 - sin(x)) = 1/(1 - x*G(0)) where G(k)= 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))).
E.g.f.: tan(x) + sec(x) = 1 + 2*x/(U(0)-x) where U(k) = 4k+2 - x^2/U(k+1).
E.g.f.: tan(x) + sec(x) = 1 + 2*x/(2*U(0)-x) where U(k) = 4*k+1 - x^2/(16*k+12 - x^2/U(k+1)).
E.g.f.: tan(x) + sec(x) = 4/(2-x*G(0))-1 where G(k) = 1 - x^2/(x^2 - 4*(2*k+1)*(2*k+3)/G(k+1)).
G.f.: 1 + x/Q(0), 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)).
G.f.: conjecture: 1 + T(0)*x/(1-x), 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)).
E.g.f.: 1+ 4*x/(T(0) - 2*x), where T(k) = 4*(2*k+1) - 4*x^2/T(k+1):
E.g.f.: T(0)-1, where T(k) = 2 + x/(4*k+1 - x/(2 - x/( 4*k+3 + x/T(k+1)))). (End)
E.g.f.: tan(x/2 + Pi/4). - Vaclav Kotesovec, Nov 08 2013
Asymptotic expansion: 4*(2*n/(Pi*e))^(n+1/2)*exp(1/2+1/(12*n) -1/(360*n^3) + 1/(1260*n^5) - ...). (See the Luschny link.) - Peter Luschny, Jul 14 2015
From Peter Bala, Sep 10 2015: (Start)
The e.g.f. A(x) = tan(x) + sec(x) satisfies A''(x) = A(x)*A'(x), hence the recurrence a(0) = 1, a(1) = 1, else a(n) = Sum_{i = 0..n-2} binomial(n-2,i)*a(i)*a(n-1-i).
Note, the same recurrence, but with the initial conditions a(0) = 0 and a(1) = 1, produces the sequence [0,1,0,1,0,4,0,34,0,496,...], an aerated version of A002105. (End)
a(n) = A186365(n)/n for n >= 1. - Anton Zakharov, Aug 23 2016
From Peter Luschny, Oct 27 2017: (Start)
a(n) = abs(2*4^n*(H(((-1)^n - 3)/8, -n) - H(((-1)^n - 7)/8, -n))) where H(z, r) are the generalized harmonic numbers.
a(n) = (-1)^binomial(n + 1, 2)*2^(2*n + 1)*(zeta(-n, 1 + (1/8)*(-7 + (-1)^n)) - zeta(-n, 1 + (1/8)*(-3 + (-1)^n))). (End)
a(n) = i*(i^n*Li_{-n}(-i) - (-i)^n*Li_{-n}(i)), where i is the imaginary unit and Li_{s}(z) is the polylogarithm. - Peter Luschny, Aug 28 2020
Sum_{n>=0} 1/a(n) = A340315. - Amiram Eldar, May 29 2021
a(n) = n!*Re([x^n](1 + I^(n^2 - n)*(2 - 2*I)/(exp(x) + I))). - Peter Luschny, Aug 09 2021

Edited by M. F. Hasler, Apr 04 2013

Title corrected by Geoffrey Critzer, May 18 2013

A001250 Number of alternating permutations of order n.

Original entry on oeis.org

1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, 707584, 5405530, 44736512, 398721962, 3807514624, 38783024290, 419730685952, 4809759350882, 58177770225664, 740742376475050, 9902996106248192, 138697748786275802, 2030847773013704704, 31029068327114173810

Offset: 0

N. J. A. Sloane

For n>1, a(n) is the number of permutations of order n with the length of longest run equal 2.
Boustrophedon transform of the Euler numbers (A000111). [Berry et al., 2013] - N. J. A. Sloane, Nov 18 2013
Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i >= j < k or i < j >= k. a(4) = 10: 0010, 0011, 0020, 0021, 0022, 0101, 0102, 0103, 0112, 0113. - Alois P. Heinz, Oct 16 2019

			1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 32*x^5 + 122*x^6 + 544*x^7 + 2770*x^8 + ...
From _Gus Wiseman_, Jun 21 2021: (Start)
The a(0) = 1 through a(4) = 10 permutations:
  ()  (1)  (1,2)  (1,3,2)  (1,3,2,4)
           (2,1)  (2,1,3)  (1,4,2,3)
                  (2,3,1)  (2,1,4,3)
                  (3,1,2)  (2,3,1,4)
                           (2,4,1,3)
                           (3,1,4,2)
                           (3,2,4,1)
                           (3,4,1,2)
                           (4,1,3,2)
                           (4,2,3,1)
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..500 (terms n=1..100 from Max Alekseyev)
Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv:1205.4581 [math.CO], 2012-2013.
Désiré André, Sur les permutations alternées, J. Math. Pur. Appl., 7 (1881), 167-184.
Désiré André, Étude sur les maxima, minima et séquences des permutations, Ann. Sci. Ecole Norm. Sup., 3, no. 1 (1884), 121-135.
Désiré André, Mémoire sur les permutations quasi-alternées, Journal de mathématiques pures et appliquées 5e série, tome 1 (1895), 315-350.
Désiré André, Mémoire sur les séquences des permutations circulaires, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
Stefano Barbero, Umberto Cerruti, and Nadir Murru, Some combinatorial properties of the Hurwitz series ring arXiv:1710.05665 [math.NT], 2017.
D. Berry, J. Broom, D. Dixon, and A. Flaherty, Umbral Calculus and the Boustrophedon Transform, 2013.
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
C. Davis, Problem 4755, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534.
Chandler Davis, Problem 4755: A Permutation Problem, Amer. Math. Monthly, 64 (1957) 596; solution by W. J. Blundon, 65 (1958), 533-534. [Denoted by P_n in solution.] [Annotated scanned copy]
S. Kitaev, Multi-avoidance of generalized patterns, Discrete Math., 260 (2003), 89-100. (See p. 100.)
S. T. Thompson, Problem E754: Skew Ordered Sequences, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Alternating Permutation

Cf. A000111. A diagonal of A010094.
Cf. A001251, A001252, A001253, A010026, A211318, A260786.
The version for permutations of prime indices is A345164.
The version for compositions is A025047, ranked by A345167.
The version for patterns is A345194.
A049774 counts permutations avoiding adjacent (1,2,3).
A344614 counts compositions avoiding adjacent (1,2,3) and (3,2,1).
A344615 counts compositions avoiding the weak adjacent pattern (1,2,3).
A344654 counts partitions without a wiggly permutation, ranked by A344653.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions, ranked by A345168.
Cf. A000041, A003242, A032020, A056986, A261962, A325534, A325535, A335452, A344652, A344740, A345165.
Row sums of A104345.

a001250 n = if n == 1 then 1 else 2 * a000111 n
-- Reinhard Zumkeller, Sep 17 2014

# With Eulerian polynomials:
A := (n, x) -> `if`(n<2, 1/2/(1+I)^(1-n), add(add((-1)^j*binomial(n+1, j)*(m+1-j)^n, j=0..m)*x^m, m=0..n-1)):
A001250 := n -> 2*(I-1)^(1-n)*exp(I*(n-1)*Pi/2)*A(n,I);
seq(A001250(i), i=0..22); # Peter Luschny, May 27 2012
# second Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> `if`(n<2, 1, 2)*b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 29 2015

a[n_] := 4*Abs[PolyLog[-n, I]]; a[0] = a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 09 2016, after M. F. Hasler *)
Table[Length[Select[Permutations[Range[n]],And@@(!(OrderedQ[#]||OrderedQ[Reverse[#]])&/@Partition[#,3,1])&]],{n,8}] (* Gus Wiseman, Jun 21 2021 *)
a[0]:=1; a[1]:=1; a[n_]:=a[n]=1/(n (n-1)) Sum[a[n-1-k] a[k] k, {k,1, n-1}]; Join[{a[0], a[1]}, Map[2 #! a[#]&, Range[2,24]]] (* Oliver Seipel, May 27 2024 *)

{a(n) = local(v=[1], t); if( n<0, 0, for( k=2, n+3, t=0; v = vector( k, i, if( i>1, t += v[k+1 - i]))); v[3])} /* Michael Somos, Feb 03 2004 */

{a(n) = if( n<0, 0, n! * polcoeff( (tan(x + x * O(x^n)) + 1 / cos(x + x * O(x^n)))^2, n))} /* Michael Somos, Feb 05 2011 */

A001250(n)=sum(m=0,n\2,my(k);(-1)^m*sum(j=0,k=n+1-2*m,binomial(k,j)*(-1)^j*(k-2*j)^(n+1))/k>>k)*2-(n==1)  \\ M. F. Hasler, May 19 2012

A001250(n)=4*abs(polylog(-n,I))-(n==1)  \\ M. F. Hasler, May 20 2012

my(x='x+O('x^66), egf=1+2*(tan(x)+1/cos(x))-2-x); Vec(serlaplace(egf)) /* Joerg Arndt, May 28 2012 */

from itertools import accumulate, islice
def A001250_gen(): # generator of terms
    yield from (1,1)
    blist = (0,2)
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
A001250_list = list(islice(A001250_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

from sympy import bernoulli, euler
def A001250(n): return 1 if n<2 else abs(((1<Chai Wah Wu, Nov 13 2024

# Algorithm of L. Seidel (1877)
def A001250_list(n) :
    R = [1]; A = {-1:0, 0:2}; k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) : Am += A[k]; A[k] = Am; k += e
        if i > 1 : R.append(A[-i//2] if i%2 == 0 else A[i//2])
    return R
A001250_list(22) # Peter Luschny, Mar 31 2012

a(n) = coefficient of x^(n-1)/(n-1)! in power series expansion of (tan(x) + sec(x))^2 = (tan(x)+1/cos(x))^2.
a(n) = coefficient of x^n/n! in power series expansion of 2*(tan(x) + sec(x)) - 2 - x. - Michael Somos, Feb 05 2011
For n>1, a(n) = 2 * A000111(n). - Michael Somos, Mar 19 2011
a(n) = 4*|Li_{-n}(i)| - [n=1] = Sum_{m=0..n/2} (-1)^m*2^(1-k)*Sum_{j=0..k} binomial(k,j)*(-1)^j*(k-2*j)^(n+1)/k - [n=1], where k = k(m) = n+1-2*m and [n=1] equals 1 if n=1 and zero else; Li denotes the polylogarithm (and i^2 = -1). - M. F. Hasler, May 20 2012
From Sergei N. Gladkovskii, Jun 18 2012: (Start)
Let E(x) = 2/(1-sin(x))-1 (essentially the e.g.f.), then
E(x) = -1 + 2*(-1/x + 1/(1-x)/x - x^3/((1-x)*((1-x)*G(0) + x^2))) where G(k) = (2*k+2)*(2*k+3)-x^2+(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction, Euler's 1st kind, 1-step).
E(x) = -1 + 2*(-1/x + 1/(1-x)/x - x^3/((1-x)*((1-x)*G(0) + x^2))) where G(k) = 8*k + 6 - x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction, Euler's 2nd kind, 2-step).
E(x) = (tan(x) + sec(x))^2 = -1 + 2/(1-x*G(0)) where G(k) = 1 - x^2/(2*(2*k+1)*(4*k+3) - 2*x^2*(2*k+1)*(4*k+3)/(x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction, 3rd kind, 3-step).
(End)
G.f.: conjecture: 2*T(0)/(1-x) -1, 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))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
a(n) ~ 2^(n+3) * n! / Pi^(n+1). - Vaclav Kotesovec, Sep 06 2014
a(n) = Sum_{k=0..n-1} A109449(n-1,k)*A000111(k). - Reinhard Zumkeller, Sep 17 2014

Edited by Max Alekseyev, May 04 2012

a(0)=1 prepended by Alois P. Heinz, Nov 29 2015

A000667 Boustrophedon transform of all-1's sequence.

Original entry on oeis.org

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Offset: 0

N. J. A. Sloane and Simon Plouffe

Fill in a triangle, like Pascal's triangle, beginning each row with a 1 and filling in rows alternately right to left and left to right.

Row sums of triangle A109449. - Reinhard Zumkeller, Nov 04 2013

			...............1..............
............1..->..2..........
.........4..<-.3...<-..1......
......1..->.5..->..8...->..9..

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Absolute value of pairwise sums of A009337.
Column k=1 of A292975.
Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.

a000667 n = if x == 1 then last xs else x
            where xs@(x:_) = a227862_row n
-- Reinhard Zumkeller, Nov 01 2013

With[{nn=30},CoefficientList[Series[Exp[x](Tan[x]+Sec[x]),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 28 2011 *)
t[, 0] = 1; t[n, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
a[n_] := t[n, n];
Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ Joerg Arndt, Jul 30 2016

from itertools import islice, accumulate
def A000667_gen(): # generator of terms
    blist = tuple()
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=1)))[-1]
A000667_list = list(islice(A000667_gen(),20)) # Chai Wah Wu, Jun 11 2022

# Algorithm of L. Seidel (1877)
def A000667_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # print [A[z] for z in (-i//2..i//2)]
        R.append(A[e*i//2])
    return R
A000667_list(10)  # Peter Luschny, Jun 02 2012

E.g.f.: exp(x) * (tan(x) + sec(x)).
Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi; lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1 + 1/(n-1). - Gerald McGarvey, Aug 13 2004
a(n) = Sum_{k=0..n} binomial(n, k)*A000111(n-k). a(2*n) = A000795(n) + A009747(n), a(2*n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005
a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013
G.f.: E(0)*x/(1-x)/(1-2*x) + 1/(1-x), where E(k) = 1 - x^2*(k + 1)*(k + 2)/(x^2*(k + 1)*(k + 2) - 2*(x*(k + 2) - 1)*(x*(k + 3) - 1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

A119879 Exponential Riordan array (sech(x),x).

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 5, 0, -6, 0, 1, 0, 25, 0, -10, 0, 1, -61, 0, 75, 0, -15, 0, 1, 0, -427, 0, 175, 0, -21, 0, 1, 1385, 0, -1708, 0, 350, 0, -28, 0, 1, 0, 12465, 0, -5124, 0, 630, 0, -36, 0, 1, -50521, 0, 62325, 0, -12810, 0, 1050, 0, -45, 0, 1

Offset: 0

Paul Barry, May 26 2006

Row sums have e.g.f. exp(x)*sech(x) (signed version of A009006). Inverse of masked Pascal triangle A119467. Transforms the sequence with e.g.f. g(x) to the sequence with e.g.f. g(x)*sech(x).
Coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows). Another version in A153641. - Philippe Deléham, Oct 26 2013
Relations to Green functions and raising/creation and lowering/annihilation/destruction operators are presented in Hodges and Sukumar and in Copeland's discussion of this sequence and 2020 pdf. - Tom Copeland, Jul 24 2020

			Triangle begins:
     1;
     0,    1;
    -1,    0,     1;
     0,   -3,     0,   1;
     5,    0,    -6,   0,   1;
     0,   25,     0, -10,   0,   1;
   -61,    0,    75,   0, -15,   0,   1;
     0, -427,     0, 175,   0, -21,   0,  1;
  1385,    0, -1708,   0, 350,   0, -28,  0,  1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 28.
Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera, 2015.
Tom Copeland, Discussion of this sequence
Tom Copeland, SKipping over Dimensions, Juggling Zeros in the Matrix, 2020.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414.
Peter Luschny, Additive decompositions of classical numbers via the Swiss-Knife polynomials [_Tom Copeland_, Oct 20 2015]
Miguel Méndez and Rafael Sánchez, On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
Miguel A. Méndez and Rafael Sánchez Lamoneda, Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.

Row sums are A155585. - Johannes W. Meijer, Apr 20 2011
Rows reversed: A081658.
Cf. A109449, A153641, A162660. - Philippe Deléham, Oct 26 2013
Cf. A000182, A046802, A119467, A133314.
Cf. A019538, A028246, A090582, A123125, A130850, A248727.

T := (n,k) -> binomial(n,k)*2^(n-k)*euler(n-k,1/2): # Peter Luschny, Jan 25 2009

T[n_, k_] := Binomial[n, k] 2^(n-k) EulerE[n-k, 1/2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Peter Luschny *)

{T(n,k) = binomial(n,k)*2^(n-k)*(2/(n-k+1))*(subst(bernpol(n-k+1, x), x, 1/2) - 2^(n-k+1)*subst(bernpol(n-k+1, x), x, 1/4))};
for(n=0,5, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 25 2019

@CachedFunction
def A119879_poly(n, x) :
    return 1 if n == 0  else add(A119879_poly(k, 0)*binomial(n, k)*(x^(n-k)-1+n%2) for k in range(n)[::2])
def A119879_row(n) :
    R = PolynomialRing(ZZ, 'x')
    return R(A119879_poly(n,x)).coeffs()  # Peter Luschny, Jul 16 2012
# Alternatively:

# uses[riordan_array from A256893]
riordan_array(sech(x), x, 9, exp=true) # Peter Luschny, Apr 19 2015

Number triangle whose k-th column has e.g.f. sech(x)*x^k/k!.
T(n,k) = C(n,k)*2^(n-k)*E_{n-k}(1/2) where C(n,k) is the binomial coefficient and E_{m}(x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
The coefficients in ascending order of x^i of the polynomials p{0}(x) = 1 and p{n}(x) = Sum_{k=0..n-1; k even} binomial(n,k)*p{k}(0)*((n mod 2) - 1 + x^(n-k)). - Peter Luschny, Jul 16 2012
E.g.f.: exp(x*z)/cosh(x). - Peter Luschny, Aug 01 2012
Sum_{k=0..n} T(n,k)*x^k = A122045(n), A155585(n), A119880(n), A119881(n) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 27 2013
With all offsets 0, let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of A248727 (the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of this entry, A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020
Triangle equals P*((I + P^2)/2)^(-1), where P denotes Pascal's triangle A007318. - Peter Bala, Mar 07 2024

A145876 Triangle read by rows: T(n,k) is the number of permutations of [n] having k-1 alternating descents (1<=k<=n). The index i is an alternating descent of a permutation p if either i is odd and p(i)>p(i+1), or i is even and p(i)

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 7, 7, 5, 16, 26, 36, 26, 16, 61, 117, 182, 182, 117, 61, 272, 594, 1056, 1196, 1056, 594, 272, 1385, 3407, 6669, 8699, 8699, 6669, 3407, 1385, 7936, 21682, 46348, 67054, 76840, 67054, 46348, 21682, 7936, 50521, 151853, 350240, 556952, 704834, 704834, 556952, 350240, 151853, 50521

Offset: 1

Emeric Deutsch, Oct 22 2008

Row sums are the factorials (A000142).
T(n,1) = T(n,n) = A000111(n) (Euler or up-down numbers).
Sum(k*T(n,k), k=1..n) = (n+1)!/2 = A001710(n+1).
From Peter Bala, Jun 11 2011: (Start)
Koutras has defined generalized Eulerian numbers associated with a sequence of polynomials - the ordinary Eulerian numbers A008292 being associated with the sequence of monomials x^n. The present array is the triangle of Eulerian numbers associated with the sequence of zigzag polynomials Z(n,x) defined in A147309.
See A109449, A147315 and A185424 for the respective analogs of the Pascal triangle, the Stirling numbers of the second kind and the Bernoulli numbers, associated with the sequence of zigzag polynomials. (End)
From Vaclav Kotesovec, Apr 29 2018: (Start)
In general, for fixed k>=1, T(n,k) ~ (4-Pi)^(k-1) * 2^(n+2) * n^(k-1) * n! / ((k-1)! * Pi^(n + k)).
Equivalently, for fixed k>=1, T(n,k) ~ (4-Pi)^(k-1) * 2^(n + 5/2) * n^(n + k - 1/2) / ((k-1)! * Pi^(n + k - 1/2) * exp(n)). (End)

			T(4,3) = 7 because we have 1243, 4123, 1342, 3124, 2134, 2341 and 4321. For example, for p=1342 the alternating descent is {2,3}; indeed, 2 is even and p(2)=3 < p(3)=4, while 3 is odd and p(3)=4 > p(4)=2.
Triangle starts:
     1;
     1,    1;
     2,    2,    2;
     5,    7,    7,    5;
    16,   26,   36,   26,   16;
    61,  117,  182,  182,  117,   61;
   272,  594, 1056, 1196, 1056,  594,  272;
  1385, 3407, 6669, 8699, 8699, 6669, 3407, 1385;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..141, flattened
D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.
M. V. Koutras, Eulerian numbers associated with sequences of polynomials, The Fibonacci Quarterly, 32 (1994), 44-57.
Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374 [math.CO], 2021.
S.-M. Ma and Y.-M. Yeh, Enumeration of permutations by number of alternating descents, Discr. Math., 339 (2016), 1362-1367.

Cf. A000142, A001710, A008292, A109449, A147315, A185424.
Columns k=1-10 give: A000111, A302691, A302895, A302896, A302897, A302898, A302899, A302900, A302901, A302902.
Cf. A302903 (T(2n+1,n+1)), A302904 (T(2n,n)), A302905 (T(n,ceiling(n/2))).

F:=t*(1-tan(u*(t-1))-sec(u*(t-1)))/(tan(u*(t-1))+sec(u*(t-1))-t): Fser:= simplify(series(F,u=0,12)): for n from 0 to 10 do P[n]:=sort(expand(factorial(n)*coeff(Fser,u,n))) end do: for n to 10 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..12);  # Alois P. Heinz, Nov 18 2013, Apr 15 2018

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[
     Sum[b[u+j-1, o-j, !t]*If[t, 1, x], {j, 1, o}] +
     Sum[b[u-j, o+j-1, !t]*If[t, x, 1], {j, 1, u}]]];
T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[0, n, True]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

E.g.f.: F(t,u) = t*(1-tan(u*(t-1))-sec(u*(t-1)))/(tan(u*(t-1))+sec(u*(t-1))-t).
From Peter Bala, Jun 11 2011: (Start)
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(n+1,k-j)*Z(n,j), where Z(n,x) are the zigzag polynomials defined in A147309.
Let M denote the triangular array A109449. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of the present array.
(End)
From Vladimir Shevelev, Jul 01 2011: (Start)
a(2^(2*n-1)-2^(n-1)+1) == 1 (mod 2^n).
If n is odd prime, then a(2*n^2-n+1) == 1 (mod 2*n) and a((n^2-n+2)/2) == (-1)^((n-1)/2).
(End)

A230960 Boustrophedon transform of factorials, cf. A000142.

Original entry on oeis.org

1, 2, 5, 17, 73, 381, 2347, 16701, 134993, 1222873, 12279251, 135425553, 1627809401, 21183890469, 296773827547, 4453511170517, 71275570240417, 1211894559430065, 21816506949416611, 414542720924028441, 8291224789668806345, 174120672081098057341

Offset: 0

Reinhard Zumkeller, Nov 05 2013

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon_transform
Index entries for sequences related to boustrophedon transform

Cf. A000142, A092580, A109449, A230961.

a230960 n = sum $ zipWith (*) (a109449_row n) a000142_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] k!, {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 02 2019 *)

from itertools import count, islice, accumulate
def A230960_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= i
A230960_list = list(islice(A230960_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000142(k).
E.g.f.: (tan(x)+sec(x))/(1-x) = (1- 12*x/(Q(0)+6*x+3*x^2))/(1-x), where Q(k) = 2*(4*k+1)*(32*k^2+16*k-x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) ~ n! * (1+sin(1))/cos(1). - Vaclav Kotesovec, Jun 12 2015
a(n) = Sum_{k=0..n} (k+1) * A092580(n,k). - Alois P. Heinz, Apr 27 2023

A062272 Boustrophedon transform of (n+1) mod 2.

Original entry on oeis.org

1, 1, 2, 5, 12, 41, 152, 685, 3472, 19921, 126752, 887765, 6781632, 56126201, 500231552, 4776869245, 48656756992, 526589630881, 6034272215552, 72989204937125, 929327412759552, 12424192360405961, 174008703107274752

Offset: 0

Frank Ellermann, Jun 16 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

A000734 (binomial transform), a(2n+1)= A003719(n), a(2n)= A000795(n),
Cf. A062161 (n mod 2).
Row sums of A162170 minus A000035. - Mats Granvik, Jun 27 2009
Cf. A059841.

a062272 n = sum $ zipWith (*) (a109449_row n) $ cycle [1,0]
-- Reinhard Zumkeller, Nov 03 2013

s[n_] = Mod[n+1, 2]; t[n_, 0] := s[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, islice
def A062272_gen(): # generator of terms
    blist, m = tuple(), 0
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062272_list = list(islice(A062272_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062272_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == 1 else 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        R.append(A[e*i//2])
    return R
A062272_list(10) # Peter Luschny, Jun 02 2012

E.g.f.: (sec(x)+tan(x))cosh(x); a(n)=(A000667(n)+A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k)*A000111(n-2k). - Philippe Deléham, Aug 28 2005
a(n) = sum(A109449(n,k) * (1 - n mod 2): k=0..n). - Reinhard Zumkeller, Nov 03 2013

A000738 Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,...

Original entry on oeis.org

0, 1, 3, 8, 25, 85, 334, 1497, 7635, 43738, 278415, 1949531, 14893000, 123254221, 1098523231, 10490117340, 106851450165, 1156403632189, 13251409502982, 160286076269309, 2040825708462175, 27283829950774822, 382127363497453243, 5595206208670390323

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000045, A000687, A000744, A092073, A109449.

a000738 n = sum $ zipWith (*) (a109449_row n) a000045_list
-- Reinhard Zumkeller, Nov 03 2013

read(transforms);
with(combinat):
F:=fibonacci;
[seq(F(n),n=0..50)];
BOUS2(%);

FullSimplify[CoefficientList[Series[(2/Sqrt[5]) * E^(x/2) * (E^(Sqrt[5]/2*x)/2 - E^(-Sqrt[5]/2*x)/2) * (Sin[x]+1) / Cos[x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec after Alois P. Heinz, Oct 05 2013 *)
t[n_, 0] := Fibonacci[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import islice, accumulate
def A000738_gen(): # generator of terms
    blist, a, b = tuple(), 0, 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
        a, b = b, a+b
A000738_list = list(islice(A000738_gen(),30)) # Chai Wah Wu, Jun 11 2022

E.g.f.: (2/sqrt(5)) * exp(x/2) * sinh((sqrt(5)/2)*x) * (sin(x)+1) / cos(x). - Alois P. Heinz, Feb 08 2011
a(n) ~ 4*(exp(sqrt(5)*Pi/2)-1) * (2*n/Pi)^(n+1/2) * exp(Pi/4-n-sqrt(5)*Pi/4) / sqrt(5). - Vaclav Kotesovec, Oct 05 2013
a(n) = sum(A109449(n,k)*A000045(k): k=0..n). - Reinhard Zumkeller, Nov 03 2013

Entry revised by N. J. A. Sloane, Mar 16 2011

A000747 Boustrophedon transform of primes.

Original entry on oeis.org

2, 5, 13, 35, 103, 345, 1325, 5911, 30067, 172237, 1096319, 7677155, 58648421, 485377457, 4326008691, 41310343279, 420783672791, 4553946567241, 52184383350787, 631210595896453, 8036822912123765, 107444407853010597, 1504827158220643895, 22034062627659931905

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A000040, A109449, A230953, A230956, A230954, A230955.

a000747 n = sum $ zipWith (*) (a109449_row n) a000040_list
-- Reinhard Zumkeller, Nov 03 2013

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

from itertools import islice, count, accumulate
from sympy import prime
def A000747_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=prime(i))))[-1]
A000747_list = list(islice(A000747_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} A109449(n,k)*A000040(k+1). - Reinhard Zumkeller, Nov 03 2013

E.g.f.: (sec(x) + tan(x)) * Sum_{k>=0} prime(k+1)*x^k/k!. - Ilya Gutkovskiy, Jun 26 2018

cp's OEIS Frontend

A343845 a(n) = Sum_{k=0..floor(n/2)} A109449(n-k, k).

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A000111 Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).

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A001250 Number of alternating permutations of order n.

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A000667 Boustrophedon transform of all-1's sequence.

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A119879 Exponential Riordan array (sech(x),x).

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