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A231894 Boustrophedon transform of the Catalan numbers A000108.

%I A231894 #31 Jun 12 2022 12:01:43
%S A231894 1,3,10,37,149,648,3039,15401,84619,505500,3287256,23250514,178382427,
%T A231894 1478782490,13187788246,125958159631,1283067859947,13886218459612,
%U A231894 159124624924418,1924735353849082,24506483918914367,327627501208785322
%N A231894 Boustrophedon transform of the Catalan numbers A000108.
%H A231894 D. Berry, J. Broom, D. Dixon, A. Flaherty, <a href="https://www.math.lsu.edu/system/files/DeAngelisProject2.pdf">Umbral Calculus and the Boustrophedon Transform</a>, 2013
%H A231894 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>).
%F A231894 a(n) = Sum_{k=0..n} A109449(n,k)*A000108(k+1). - _Philippe Deléham_, Nov 20 2013
%F A231894 E.g.f.: exp(2*x)*I_1(2*x)*(sec(x)+tan(x))/x, where I_1(2*x) is the modified Bessel function of the first kind. - _Sergei N. Gladkovskii_, Nov 19 2014
%F A231894 a(n) ~ n! * exp(Pi) * BesselI(1, Pi) * 2^(n+3) / Pi^(n+2). - _Vaclav Kotesovec_, Jun 12 2015
%e A231894 G.f. = 1 + 3*x + 10*x^2 +37*x^3 + 149*x^4 + 648*x^5 + 3039*x^6 + 15401*x^7 + ...
%p A231894 A000111 := proc(n)
%p A231894     option remember;
%p A231894     sec(x)+tan(x) ;
%p A231894     coeftayl(%,x=0,n)*n! ;
%p A231894 end proc:
%p A231894 A109449 := proc(n,k)
%p A231894     binomial(n,k)*A000111(n-k) ;
%p A231894 end proc:
%p A231894 A231894 := proc(n)
%p A231894     add( A109449(n,k)*A000108(k+1),k=0..n) ;
%p A231894 end proc:
%p A231894 seq(A231894(n),n=0..30) ; # _R. J. Mathar_, Oct 04 2014
%o A231894 (Python)
%o A231894 from itertools import accumulate, count, islice
%o A231894 def A231894_gen(): # generator of terms
%o A231894     blist, c = tuple(), 1
%o A231894     for i in count(1):
%o A231894         yield (blist := tuple(accumulate(reversed(blist),initial=c)))[-1]
%o A231894         c = c*(4*i+2)//(i+2)
%o A231894 A231894_list = list(islice(A231894_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A231894 Cf. A000108, A000736, A000753, A109449.
%K A231894 nonn
%O A231894 0,2
%A A231894 _N. J. A. Sloane_, Nov 18 2013