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A000734 Boustrophedon transform of 1,1,2,4,8,16,32,...

%I A000734 #44 Jun 12 2022 11:49:24
%S A000734 1,2,5,15,49,177,715,3255,16689,95777,609875,4270695,32624329,
%T A000734 269995377,2406363835,22979029335,234062319969,2533147494977,
%U A000734 29027730898595,351112918079175,4470508510495609,59766296291090577
%N A000734 Boustrophedon transform of 1,1,2,4,8,16,32,...
%C A000734 Binomial transform of A062272. - _Paul Barry_, Jan 21 2005
%H A000734 Reinhard Zumkeller, <a href="/A000734/b000734.txt">Table of n, a(n) for n = 0..400</a>
%H A000734 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A000734 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (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 A000734 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000734 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A000734 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000734 E.g.f.: (1 + exp(2*x))*(sec(x) + tan(x))/2. - _Paul Barry_, Jan 21 2005
%F A000734 a(n) ~ n! * (1 + exp(Pi)) * (2/Pi)^(n+1). - _Vaclav Kotesovec_, Oct 07 2013
%t A000734 CoefficientList[Series[(1+E^(2*x))*(Sec[x]+Tan[x])/2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 07 2013 *)
%t A000734 t[n_, 0] := If[n == 0, 1, 2^(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 A000734 (Sage) # Algorithm of L. Seidel (1877)
%o A000734 def A000734_list(n) :
%o A000734     A = {-1:0, 0:1}; R = []
%o A000734     k = 0; e = 1; Bm = 1
%o A000734     for i in range(n) :
%o A000734         Am = Bm
%o A000734         A[k + e] = 0
%o A000734         e = -e
%o A000734         for j in (0..i) :
%o A000734             Am += A[k]
%o A000734             A[k] = Am
%o A000734             k += e
%o A000734         Bm += Bm
%o A000734         R.append(A[e*i//2]/2)
%o A000734     return R
%o A000734 A000734_list(22) # _Peter Luschny_, Jun 02 2012
%o A000734 (Haskell)
%o A000734 a000734 n = sum $ zipWith (*) (a109449_row n) (1 : a000079_list)
%o A000734 -- _Reinhard Zumkeller_, Nov 04 2013
%o A000734 (Python)
%o A000734 from itertools import count, accumulate, islice
%o A000734 def A000734_gen(): # generator of terms
%o A000734     yield 1
%o A000734     blist, m = (1,), 1
%o A000734     while True:
%o A000734         yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
%o A000734         m *= 2
%o A000734 A000734_list = list(islice(A000734_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A000734 Cf. A109449, A000079, A000752.
%K A000734 nonn
%O A000734 0,2
%A A000734 _N. J. A. Sloane_, _Simon Plouffe_