cp's OEIS Frontend

A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...

%I A000736 #47 Jun 12 2022 11:49:03
%S A000736 1,2,4,10,32,120,513,2455,13040,76440,492231,3465163,26530503,
%T A000736 219754535,1959181266,18710532565,190588702776,2062664376064,
%U A000736 23636408157551,285900639990875,3640199365715769,48665876423760247
%N A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...
%H A000736 Reinhard Zumkeller, <a href="/A000736/b000736.txt">Table of n, a(n) for n = 0..400</a>
%H A000736 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H A000736 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (<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 A000736 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A000736 Wikipedia, <a href="https://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H A000736 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A000736 E.g.f.: (sec(x) + tan(x))*(integral(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)),x)+1). - _Sergei N. Gladkovskii_, Oct 30 2014
%F A000736 a(n) ~ n! * (6/Pi+2*exp(Pi)*((2-1/Pi)*BesselI(0,Pi)-2*BesselI(1,Pi))) * 2^n / Pi^n. - _Vaclav Kotesovec_, Oct 30 2014
%p A000736 egf := (sec(x/2)+tan(x/2))*(exp(x)*((x-1/2)*BesselI(0,x)-x*BesselI(1,x))+3/2);
%p A000736 s := n -> 2^n*n!*coeff(series(egf,x,n+2),x,n); seq(s(n), n=0..22); # _Peter Luschny_, Oct 30 2014, after _Sergei N. Gladkovskii_
%t A000736 CoefficientList[Series[1/2*(3 + E^(2*x)*((4*x-1)*BesselI[0, 2*x] - 4*x*BesselI[1, 2*x]))*(Sec[x] + Tan[x]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 30 2014, after _Peter Luschny_ *)
%t A000736 t[n_, 0] := If[n == 0, 1, CatalanNumber[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 A000736 (Haskell)
%o A000736 a000736 n = sum $ zipWith (*) (a109449_row n) (1 : a000108_list)
%o A000736 -- _Reinhard Zumkeller_, Nov 05 2013
%o A000736 (Python)
%o A000736 from itertools import accumulate, count, islice
%o A000736 def A000736_gen(): # generator of terms
%o A000736     yield 1
%o A000736     blist, c = (1,), 1
%o A000736     for i in count(0):
%o A000736         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
%o A000736         c = c*(4*i+2)//(i+2)
%o A000736 A000736_list = list(islice(A000736_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A000736 Cf. A000108, A000753, A109449.
%K A000736 nonn
%O A000736 0,2
%A A000736 _N. J. A. Sloane_, _Simon Plouffe_