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%I #52 Jun 12 2022 12:01:58
%S 1,2,5,14,42,144,563,2526,12877,73778,469616,3288428,25121097,
%T 207902202,1852961189,17694468210,180234349762,1950592724756,
%U 22352145975707,270366543452702,3442413745494957,46021681757269830
%N Boustrophedon transform (second version) of Fibonacci numbers 1,1,2,3,...
%H Reinhard Zumkeller, <a href="/A000744/b000744.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 Ser. A, 76(1) (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 J. Millar, N. J. A. Sloane and N. E. Young, <a href="https://doi.org/10.1006/jcta.1996.0087">A new operation on sequences: the Boustrophedon transform</a>, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
%H Ludwig Seidel, <a href="https://babel.hathitrust.org/cgi/pt?id=hvd.32044092897461&view=1up&seq=175">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the <a href="https://www.hathitrust.org/accessibility">HATHI TRUST Digital Library</a>]
%H Ludwig Seidel, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1877_0157-0187.pdf">Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through <a href="https://de.wikipedia.org/wiki/ZOBODAT">ZOBODAT</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)*A000045(k+1). - _Reinhard Zumkeller_, Nov 03 2013
%F E.g.f.: (1/10)*(sec(x)+tan(x))*((5^(1/2)+1)*exp(1/2*x*(5^(1/2)+1))+(5^(1/2)-1)*exp(1/2*x*(-5^(1/2)+1)))*5^(1/2). - _Sergei N. Gladkovskii_, Oct 30 2014
%F a(n) ~ n! * (sqrt(5) - 1 + (1+sqrt(5)) * exp(sqrt(5)*Pi/2)) * 2^(n+1) / (sqrt(5) * exp((sqrt(5)-1)*Pi/4) * Pi^(n+1)). - _Vaclav Kotesovec_, Jun 12 2015
%e G.f. = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 144*x^5 + 563*x^6 + 2526*x^7 + ...
%p read(transforms);
%p with(combinat):
%p F:=fibonacci;
%p [seq(F(n), n=1..50)];
%p BOUS2(%);
%t s[k_] := SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, k}] k!;
%t b[n_, k_] := Binomial[n, k] s[n - k];
%t a[n_] := Sum[b[n, k] Fibonacci[k + 1], {k, 0, n}];
%t Array[a, 22, 0] (* _Jean-François Alcover_, Jun 01 2019 *)
%o (Haskell)
%o a000744 n = sum $ zipWith (*) (a109449_row n) $ tail a000045_list
%o -- _Reinhard Zumkeller_, Nov 03 2013
%o (Python)
%o from itertools import accumulate, islice
%o def A000744_gen(): # generator of terms
%o blist, a, b = tuple(), 1, 1
%o while True:
%o yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
%o a, b = b, a+b
%o A000744_list = list(islice(A000744_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y Cf. A000045, A000687, A000738, A092073.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Entry revised by _N. J. A. Sloane_, Mar 16 2011
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