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A000667 Boustrophedon transform of all-1's sequence.

%I A000667 #107 Jan 05 2025 19:51:31
%S A000667 1,2,4,9,24,77,294,1309,6664,38177,243034,1701909,13001604,107601977,
%T A000667 959021574,9157981309,93282431344,1009552482977,11568619292914,
%U A000667 139931423833509,1781662223749884,23819069385695177,333601191667149054,4884673638115922509
%N A000667 Boustrophedon transform of all-1's sequence.
%C A000667 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 A000667 Row sums of triangle A109449. - _Reinhard Zumkeller_, Nov 04 2013
%H A000667 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 A000667 C. K. Cook, M. R. Bacon, and R. A. Hillman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-3/CookBaconHillman01222017.pdf">Higher-order Boustrophedon transforms for certain well-known sequences</a>, Fib. Q., 55(3) (2017), 201-208.
%H A000667 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A000667 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 (<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 A000667 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 A000667 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 A000667 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 A000667 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 A000667 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000667 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A000667 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>.
%F A000667 E.g.f.: exp(x) * (tan(x) + sec(x)).
%F A000667 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
%F A000667 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
%F A000667 a(n) = A227862(n, n * (n mod 2)). - _Reinhard Zumkeller_, Nov 01 2013
%F A000667 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 A000667 a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Jun 12 2015
%e A000667 ...............1..............
%e A000667 ............1..->..2..........
%e A000667 .........4..<-.3...<-..1......
%e A000667 ......1..->.5..->..8...->..9..
%t A000667 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 A000667 t[_, 0] = 1; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
%t A000667 a[n_] := t[n, n];
%t A000667 Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)
%o A000667 (Sage) # Algorithm of L. Seidel (1877)
%o A000667 def A000667_list(n) :
%o A000667     R = []; A = {-1:0, 0:0}
%o A000667     k = 0; e = 1
%o A000667     for i in range(n) :
%o A000667         Am = 1
%o A000667         A[k + e] = 0
%o A000667         e = -e
%o A000667         for j in (0..i) :
%o A000667             Am += A[k]
%o A000667             A[k] = Am
%o A000667             k += e
%o A000667         # print [A[z] for z in (-i//2..i//2)]
%o A000667         R.append(A[e*i//2])
%o A000667     return R
%o A000667 A000667_list(10)  # _Peter Luschny_, Jun 02 2012
%o A000667 (Haskell)
%o A000667 a000667 n = if x == 1 then last xs else x
%o A000667             where xs@(x:_) = a227862_row n
%o A000667 -- _Reinhard Zumkeller_, Nov 01 2013
%o A000667 (PARI) x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ _Joerg Arndt_, Jul 30 2016
%o A000667 (Python)
%o A000667 from itertools import islice, accumulate
%o A000667 def A000667_gen(): # generator of terms
%o A000667     blist = tuple()
%o A000667     while True:
%o A000667         yield (blist := tuple(accumulate(reversed(blist),initial=1)))[-1]
%o A000667 A000667_list = list(islice(A000667_gen(),20)) # _Chai Wah Wu_, Jun 11 2022
%Y A000667 Absolute value of pairwise sums of A009337.
%Y A000667 Column k=1 of A292975.
%Y A000667 Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.
%K A000667 nonn,easy,nice
%O A000667 0,2
%A A000667 _N. J. A. Sloane_ and _Simon Plouffe_