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A000737 Boustrophedon transform of natural numbers, cf. A000027.

%I A000737 #50 Jun 12 2022 12:00:59
%S A000737 1,3,8,21,60,197,756,3367,17136,98153,624804,4375283,33424512,
%T A000737 276622829,2465449252,23543304919,239810132288,2595353815825,
%U A000737 29740563986500,359735190398875,4580290700420064,61233976084442741
%N A000737 Boustrophedon transform of natural numbers, cf. A000027.
%H A000737 Reinhard Zumkeller, <a href="/A000737/b000737.txt">Table of n, a(n) for n = 0..400</a>
%H A000737 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A000737 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 A000737 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A000737 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H A000737 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000737 E.g.f.: (1 + x)*(tan x + sec x)*exp(x).
%F A000737 a(n) ~ n! * (Pi + 2)*exp(Pi/2)*2^(n+1)/Pi^(n+1). - _Vaclav Kotesovec_, Oct 02 2013
%t A000737 CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x])* E^x, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 02 2013 *)
%t A000737 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 A000737 (Sage) # Algorithm of L. Seidel (1877)
%o A000737 def A000737_list(n) :
%o A000737     R = []; A = {-1:0, 0:0}
%o A000737     k = 0; e = 1
%o A000737     for i in range(n) :
%o A000737         Am = i+1
%o A000737         A[k + e] = 0
%o A000737         e = -e
%o A000737         for j in (0..i) :
%o A000737             Am += A[k]
%o A000737             A[k] = Am
%o A000737             k += e
%o A000737         # To trace the algorithm remove the comment sign.
%o A000737         # print([A[z] for z in (-i//2..i//2)])
%o A000737         R.append(A[e*i//2])
%o A000737     return R
%o A000737 A000737_list(10) # _Peter Luschny_, Jun 02 2012
%o A000737 (Haskell)
%o A000737 a000737 n = sum $ zipWith (*) (a109449_row n) [1..]
%o A000737 -- _Reinhard Zumkeller_, Nov 05 2013
%o A000737 (Python)
%o A000737 from itertools import count, accumulate, islice
%o A000737 def A000737_gen(): # generator of terms
%o A000737     blist = tuple()
%o A000737     for i in count(1):
%o A000737         yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
%o A000737 A000737_list = list(islice(A000737_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A000737 Cf. A231179.
%K A000737 nonn
%O A000737 0,2
%A A000737 _N. J. A. Sloane_