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A262600 Number of Dyck paths of semilength n and height exactly 4.

%I A262600 #31 Aug 08 2025 01:54:44
%S A262600 0,0,0,0,1,7,33,132,484,1684,5661,18579,59917,190696,600744,1877256,
%T A262600 5828185,17998783,55342617,169552428,517884748,1577812060,4796682165,
%U A262600 14555626635,44100374341,133436026192,403279293648,1217616622992,3673214880049,11072960931319
%N A262600 Number of Dyck paths of semilength n and height exactly 4.
%H A262600 Colin Barker, <a href="/A262600/b262600.txt">Table of n, a(n) for n = 0..1000</a>
%H A262600 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-16,13,-3).
%F A262600 a(n) = A124302(n) - A001519(n).
%F A262600 G.f.: x^4/((x-1)*(3*x-1)*(x^2-3*x+1)).
%F A262600 a(n) = A080936(n,4).
%F A262600 From _Colin Barker_, Feb 08 2016: (Start)
%F A262600 a(n) = 7*a(n-1)-16*a(n-2)+13*a(n-3)-3*a(n-4) for n>4.
%F A262600 a(n) = 2^(-1-n)*(5*2^n*(3+3^n)+3*(-5+sqrt(5))*(3+sqrt(5))^n-3*(3-sqrt(5))^n*(5+sqrt(5)))/15 for n>0. (End)
%F A262600 E.g.f.: (2 + 3*exp(x) + exp(3*x))/6 - exp(3*x/2)*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2))/5. - _Stefano Spezia_, May 21 2024
%e A262600 a(4) = 1 because the only favorable path is UUUUDDDD.
%t A262600 CoefficientList[ Series[x^4/((x-1) (3 x-1) (x^2-3 x+1)), {x, 0, 30}], x]
%o A262600 (PARI) a(n) = if( n<1, n==0, (3^(n-1) + 1) / 2) - fibonacci(2*n-1); vector(30, n, a(n-1)) \\ _Altug Alkan_, Sep 25 2015
%o A262600 (Magma) [((3^(n-1)+1)/2)-Fibonacci(2*n-1): n in [1.. 35]]; // _Vincenzo Librandi_, Sep 26 2015
%o A262600 (PARI) concat(vector(4), Vec(x^4/((1-x)*(1-3*x)*(1-3*x+x^2)) + O(x^100))) \\ _Colin Barker_, Feb 08 2016
%Y A262600 Cf. A001519, A124302.
%Y A262600 Column k=4 of A080936.
%K A262600 nonn,easy
%O A262600 0,6
%A A262600 _Ran Pan_, Sep 25 2015