A000753 - OEIS

%I #37 Jun 11 2022 15:33:05

%S 1,2,5,16,59,243,1101,5461,29619,175641,1137741,8031838,61569345,

%T 510230087,4549650423,43452408496,442620720531,4790322653809,

%U 54893121512453,663974736739232,8453986695437957,113021461431438475

%N Boustrophedon transform of Catalan numbers.

%H Reinhard Zumkeller, <a href="/A000753/b000753.txt">Table of n, a(n) for n = 0..400</a>

%H 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</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)*A000108(k). - _Reinhard Zumkeller_, Nov 05 2013

%F 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 a(n) ~ n! * exp(Pi) * (BesselI(0, Pi) - BesselI(1, Pi)) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Oct 30 2014

%t 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 (Haskell)

%o a000753 n = sum $ zipWith (*) (a109449_row n) a000108_list

%o -- _Reinhard Zumkeller_, Nov 05 2013

%o (Python)

%o from itertools import accumulate, count, islice

%o def A000753_gen(): # generator of terms

%o     blist, c = tuple(), 1

%o     for i in count(0):

%o         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]

%o         c = c*(4*i+2)//(i+2)

%o A000753_list = list(islice(A000753_gen(),30)) # _Chai Wah Wu_, Jun 11 2022

%Y Cf. A000108, A000736, A109449.

%K nonn

%O 0,2

%A _N. J. A. Sloane_