A000660 - OEIS

%I #49 Jun 13 2022 03:03:00

%S 1,2,5,14,41,136,523,2330,11857,67912,432291,3027166,23125673,

%T 191389108,1705788659,16289080922,165919213089,1795666675824,

%U 20576824369027,248892651678198,3168999664907705,42366404751871660

%N Boustrophedon transform of 1,1,2,3,4,5,...

%H Reinhard Zumkeller, <a href="/A000660/b000660.txt">Table of n, a(n) for n = 0..400</a>

%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 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 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 a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - _Reinhard Zumkeller_, Nov 04 2013

%F E.g.f.:  (x*exp(x) + 1)*(sec(x) + tan(x)). - _Sergei N. Gladkovskii_, Oct 28 2014

%F a(n) = A231179(n) + A000111(n). - _Sergei N. Gladkovskii_, Oct 28 2014

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

%p 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 a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Jul 30 2018, after _Sergei N. Gladkovskii_ *)

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

%o def A000660_list(n) :

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

%o     k = 0; e = 1

%o     for i in range(n) :

%o         Am = i

%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 A000660_list(10) # _Peter Luschny_, Jun 02 2012

%o (Haskell)

%o a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])

%o -- _Reinhard Zumkeller_, Nov 04 2013

%o (Python)

%o from itertools import accumulate, count, islice

%o def A000660_gen(): # generator of terms

%o     yield 1

%o     blist = (1,)

%o     for i in count(1):

%o         yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]

%o A000660_list = list(islice(A000660_gen(),40)) # _Chai Wah Wu_, Jun 12 2022

%K nonn

%O 0,2

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