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A000753 Boustrophedon transform of Catalan numbers.

%I A000753 #37 Jun 11 2022 15:33:05
%S A000753 1,2,5,16,59,243,1101,5461,29619,175641,1137741,8031838,61569345,
%T A000753 510230087,4549650423,43452408496,442620720531,4790322653809,
%U A000753 54893121512453,663974736739232,8453986695437957,113021461431438475
%N A000753 Boustrophedon transform of Catalan numbers.
%H A000753 Reinhard Zumkeller, <a href="/A000753/b000753.txt">Table of n, a(n) for n = 0..400</a>
%H A000753 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 A000753 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A000753 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000753 a(n) = Sum_{k=0..n} A109449(n,k)*A000108(k). - _Reinhard Zumkeller_, Nov 05 2013
%F A000753 E.g.f.: (sec(x) + tan(x))*exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - _Sergei N. Gladkovskii_, Oct 30 2014
%F A000753 a(n) ~ n! * exp(Pi) * (BesselI(0, Pi) - BesselI(1, Pi)) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Oct 30 2014
%t A000753 CoefficientList[Series[E^(2*x) * (BesselI[0,2*x] - BesselI[1,2*x]) * (Sec[x] + Tan[x]),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Oct 30 2014 after _Sergei N. Gladkovskii_ *)
%o A000753 (Haskell)
%o A000753 a000753 n = sum $ zipWith (*) (a109449_row n) a000108_list
%o A000753 -- _Reinhard Zumkeller_, Nov 05 2013
%o A000753 (Python)
%o A000753 from itertools import accumulate, count, islice
%o A000753 def A000753_gen(): # generator of terms
%o A000753     blist, c = tuple(), 1
%o A000753     for i in count(0):
%o A000753         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
%o A000753         c = c*(4*i+2)//(i+2)
%o A000753 A000753_list = list(islice(A000753_gen(),30)) # _Chai Wah Wu_, Jun 11 2022
%Y A000753 Cf. A000108, A000736, A109449.
%K A000753 nonn
%O A000753 0,2
%A A000753 _N. J. A. Sloane_