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A000752 Boustrophedon transform of powers of 2.

%I A000752 #41 Jun 12 2022 12:00:48
%S A000752 1,3,9,28,93,338,1369,6238,31993,183618,1169229,8187598,62545893,
%T A000752 517622498,4613366689,44054301358,448733127793,4856429646978,
%U A000752 55650582121749,673136951045518,8570645832753693,114581094529057058,1604780986816602409,23497612049668468078
%N A000752 Boustrophedon transform of powers of 2.
%H A000752 Reinhard Zumkeller, <a href="/A000752/b000752.txt">Table of n, a(n) for n = 0..400</a>
%H A000752 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>.
%H A000752 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 A000752 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000752 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H A000752 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000752 E.g.f.: exp(2*x) (tan(x) + sec(x)).
%F A000752 a(n) = Sum_{k=0..n} A109449(n,k)*2^k. - _Reinhard Zumkeller_, Nov 03 2013
%F A000752 G.f.: E(0)*x/(1 - 2*x)/(1 - 3*x) + 1/(1 - 2*x), where E(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(x*(k+3) - 1)*(x*(k+4)  -1)/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2014
%F A000752 a(n) ~ n! * exp(Pi) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Jun 12 2015
%t A000752 t[n_, 0] := 2^n; 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 *)
%t A000752 With[{nn=30},CoefficientList[Series[Exp[2x](Tan[ x]+Sec[x]),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 15 2018 *)
%o A000752 (Haskell)
%o A000752 a000752 n = sum $ zipWith (*) (a109449_row n) a000079_list
%o A000752 -- _Reinhard Zumkeller_, Nov 03 2013
%o A000752 (Python)
%o A000752 from itertools import accumulate, islice
%o A000752 def A000752_gen(): # generator of terms
%o A000752     blist, m = tuple(), 1
%o A000752     while True:
%o A000752         yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
%o A000752         m *= 2
%o A000752 A000752_list = list(islice(A000752_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A000752 Cf. A000079, A000734.
%Y A000752 Column k=2 of A292975.
%K A000752 nonn
%O A000752 0,2
%A A000752 _N. J. A. Sloane_