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 A000667 Boustrophedon transform of all-1's sequence. 35

%I

%S 1,2,4,9,24,77,294,1309,6664,38177,243034,1701909,13001604,107601977,

%T 959021574,9157981309,93282431344,1009552482977,11568619292914,

%U 139931423833509,1781662223749884,23819069385695177,333601191667149054,4884673638115922509

%N Boustrophedon transform of all-1's sequence.

%C 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.

%C a(n) = A227862(n, n * (n mod 2)). - _Reinhard Zumkeller_, Nov 01 2013

%C Row sums of triangle A109449. - _Reinhard Zumkeller_, Nov 04 2013

%H Alois P. Heinz, <a href="/A000667/b000667.txt">Table of n, a(n) for n = 0..485</a> (first 101 terms from T. D. Noe)

%H C. K. Cook, M. R. Bacon, and R. A. Hillman, <a href="https://www.fq.math.ca/Abstracts/55-3/cook.pdf">Higher-order Boustrophedon transforms for certain well-known sequences</a>, Fib. Q., 55(3) (2017), 201-208.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.

%H J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory Ser. A, 76(1) (1996) 44-54 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).

%H J. Millar, N. J. A. Sloane, and N. E. Young, <a href="https://doi.org/10.1006/jcta.1996.0087">A new operation on sequences: the Boustrophedon transform</a>, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.

%H Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&amp;view=1up&amp;seq=176">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the <a href="https://www.hathitrust.org/accessibility">HATHI TRUST Digital Library</a>]

%H Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through <a href="https://de.wikipedia.org/wiki/ZOBODAT">ZOBODAT</a>]

%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>.

%F E.g.f.: exp(x) * (tan(x) + sec(x)).

%F Lim 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

%F a(n) = Sum_{k, k>=0} binomial(n, k)*A000111(n-k). a(2n) = A000795(n) + A009747(n), a(2n+1) = A002084(n) + A003719(n). - _Philippe Deléham_, Aug 28 2005

%F 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

%F a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Jun 12 2015

%e ...............1..............

%e ............1..->..2..........

%e .........4..<-.3...<-..1......

%e ......1..->.5..->..8...->..9..

%t 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 t[_, 0] = 1; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];

%t a[n_] := t[n, n];

%t Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)

%o (Sage) # Algorithm of L. Seidel (1877)

%o def A000667_list(n) :

%o R = []; A = {-1:0, 0:0}

%o k = 0; e = 1

%o for i in range(n) :

%o Am = 1

%o A[k + e] = 0

%o e = -e

%o for j in (0..i) :

%o Am += A[k]

%o A[k] = Am

%o k += e

%o # print [A[z] for z in (-i//2..i//2)]

%o R.append(A[e*i//2])

%o return R

%o A000667_list(10) # _Peter Luschny_, Jun 02 2012

%o a000667 n = if x == 1 then last xs else x

%o where xs@(x:_) = a227862_row n

%o -- _Reinhard Zumkeller_, Nov 01 2013

%o (PARI) x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ _Joerg Arndt_, Jul 30 2016

%Y Absolute value of pairwise sums of A009337.

%Y Column k=1 of A292975.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified September 20 10:04 EDT 2020. Contains 337264 sequences. (Running on oeis4.)