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A000737 Boustrophedon transform of natural numbers, cf. A000027. 5

%I #50 Jun 12 2022 12:00:59

%S 1,3,8,21,60,197,756,3367,17136,98153,624804,4375283,33424512,

%T 276622829,2465449252,23543304919,239810132288,2595353815825,

%U 29740563986500,359735190398875,4580290700420064,61233976084442741

%N Boustrophedon transform of natural numbers, cf. A000027.

%H Reinhard Zumkeller, <a href="/A000737/b000737.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 E.g.f.: (1 + x)*(tan x + sec x)*exp(x).

%F a(n) ~ n! * (Pi + 2)*exp(Pi/2)*2^(n+1)/Pi^(n+1). - _Vaclav Kotesovec_, Oct 02 2013

%t CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x])* E^x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 02 2013 *)

%t t[n_, 0] := 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 *)

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

%o def A000737_list(n) :

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

%o k = 0; e = 1

%o for i in range(n) :

%o Am = i+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 # To trace the algorithm remove the comment sign.

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

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

%o return R

%o A000737_list(10) # _Peter Luschny_, Jun 02 2012

%o (Haskell)

%o a000737 n = sum $ zipWith (*) (a109449_row n) [1..]

%o -- _Reinhard Zumkeller_, Nov 05 2013

%o (Python)

%o from itertools import count, accumulate, islice

%o def A000737_gen(): # generator of terms

%o blist = tuple()

%o for i in count(1):

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

%o A000737_list = list(islice(A000737_gen(),40)) # _Chai Wah Wu_, Jun 12 2022

%Y Cf. A231179.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)