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 A000736 Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ... 4

%I

%S 1,2,4,10,32,120,513,2455,13040,76440,492231,3465163,26530503,

%T 219754535,1959181266,18710532565,190588702776,2062664376064,

%U 23636408157551,285900639990875,3640199365715769,48665876423760247

%N Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ...

%H Reinhard Zumkeller, <a href="/A000736/b000736.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 on 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Wikipedia, <a href="http://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 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 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 egf := (sec(x/2)+tan(x/2))*(exp(x)*((x-1/2)*BesselI(0,x)-x*BesselI(1,x))+3/2);

%p 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 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 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 n = sum \$ zipWith (*) (a109449_row n) (1 : a000108_list)

%o -- _Reinhard Zumkeller_, Nov 05 2013

%Y Cf. A000108, A000753, A109449.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified January 19 20:11 EST 2019. Contains 319309 sequences. (Running on oeis4.)