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A225015 Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.

%I A225015 #20 Apr 04 2024 10:17:04
%S A225015 0,1,1,5,18,66,245,918,3465,13156,50193,192270,739024,2848860,
%T A225015 11009778,42642460,165480975,643281480,2504501625,9764299710,
%U A225015 38115568260,148955040300,582714871830,2281745337300,8942420595810,35074414899576,137672461877850,540756483094828
%N A225015 Number of sawtooth patterns of length 1 in all Dyck paths of semilength n.
%C A225015 A sawtooth pattern of length 1 is UD not followed by UD.
%C A225015 First differences of A024482.
%H A225015 Alois P. Heinz, <a href="/A225015/b225015.txt">Table of n, a(n) for n = 0..1000</a>
%F A225015 a(0)=0, a(1)=1, a(n) = A024482(n) - A024482(n-1) for n >= 2.
%F A225015 From _G. C. Greubel_, Apr 03 2024: (Start)
%F A225015 G.f.: (1-x)^2*(1 - sqrt(1-4*x))/(2*sqrt(1-4*x)).
%F A225015 E.g.f.: -(1/4)*(2-4*x+x^2) + (1/12)*Exp(2*x)*((6-12*x+43*x^2-24*x^3) *BesselI(0, 2*x) - 4*x*(7-5*x)*BesselI(1,2*x) - 3*x^2*(13-8*x)* BesselI(2,2*x)). (End)
%p A225015 a:= proc(n) option remember; `if`(n<4, [0, 1, 1, 5][n+1],
%p A225015        (n-1)*(3*n-4)*(4*n-10)*a(n-1)/(n*(n-2)*(3*n-7)))
%p A225015     end:
%p A225015 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 24 2013
%t A225015 Join[{0, 0, 1}, Table[(Binomial[2n, n]-Binomial[2n-2, n-1])/2, {n, 2, 25}]] // Differences (* _Jean-François Alcover_, Nov 12 2020 *)
%o A225015 (Magma)
%o A225015 A024482:= func< n | (3*n-2)*Catalan(n-1)/2 >;
%o A225015 A225015:= func< n | n le 2 select Floor((n+1)/2) else A024482(n) - A024482(n-1) >;
%o A225015 [A225015(n): n in [0..40]]; // _G. C. Greubel_, Apr 03 2024
%o A225015 (SageMath)
%o A225015 def A024482(n): return (3*n-2)*catalan_number(n-1)/2
%o A225015 def A225015(n): return floor((n+1)/2) if n<3 else A024482(n) - A024482(n-1)
%o A225015 [A225015(n) for n in range(41)] # _G. C. Greubel_, Apr 03 2024
%Y A225015 Cf. A024482, A097613.
%K A225015 nonn
%O A225015 0,4
%A A225015 _David Scambler_, Apr 23 2013