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A000660 Boustrophedon transform of 1,1,2,3,4,5,...

%I A000660 #53 Jul 06 2025 11:33:59
%S A000660 1,2,5,14,41,136,523,2330,11857,67912,432291,3027166,23125673,
%T A000660 191389108,1705788659,16289080922,165919213089,1795666675824,
%U A000660 20576824369027,248892651678198,3168999664907705,42366404751871660,593368400878431795,8688251294851280594
%N A000660 Boustrophedon transform of 1,1,2,3,4,5,...
%H A000660 Reinhard Zumkeller, <a href="/A000660/b000660.txt">Table of n, a(n) for n = 0..400</a>
%H A000660 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A000660 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 (<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 A000660 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A000660 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H A000660 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000660 a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - _Reinhard Zumkeller_, Nov 04 2013
%F A000660 E.g.f.: (x*exp(x) + 1)*(sec(x) + tan(x)). - _Sergei N. Gladkovskii_, Oct 28 2014
%F A000660 a(n) = A231179(n) + A000111(n). - _Sergei N. Gladkovskii_, Oct 28 2014
%F A000660 a(n) ~ n! * (2 + Pi*exp(Pi/2)) * (2/Pi)^(n+1). - _Vaclav Kotesovec_, Jun 12 2015
%p A000660 seq(coeff(series(factorial(n)*(x*exp(x)+1)*(sec(x)+tan(x)), x,n+1),x,n),n=0..25); # _Muniru A Asiru_, Jul 30 2018
%t A000660 a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];
%t A000660 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jul 30 2018, after _Sergei N. Gladkovskii_ *)
%o A000660 (Sage) # Algorithm of L. Seidel (1877)
%o A000660 def A000660_list(n) :
%o A000660     R = []; A = {-1:0, 0:1}
%o A000660     k = 0; e = 1
%o A000660     for i in range(n) :
%o A000660         Am = i
%o A000660         A[k + e] = 0
%o A000660         e = -e
%o A000660         for j in (0..i) :
%o A000660             Am += A[k]
%o A000660             A[k] = Am
%o A000660             k += e
%o A000660         print([A[z] for z in (-i//2..i//2)])
%o A000660         R.append(A[e*i//2])
%o A000660     return R
%o A000660 A000660_list(10) # _Peter Luschny_, Jun 02 2012
%o A000660 (Haskell)
%o A000660 a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
%o A000660 -- _Reinhard Zumkeller_, Nov 04 2013
%o A000660 (Python)
%o A000660 from itertools import accumulate, count, islice
%o A000660 def A000660_gen(): # generator of terms
%o A000660     yield 1
%o A000660     blist = (1,)
%o A000660     for i in count(1):
%o A000660         yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
%o A000660 A000660_list = list(islice(A000660_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A000660 Cf. A000111, A028310, A109449, A231179.
%K A000660 nonn
%O A000660 0,2
%A A000660 _N. J. A. Sloane_, _Simon Plouffe_