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A262600
Number of Dyck paths of semilength n and height exactly 4.
4
0, 0, 0, 0, 1, 7, 33, 132, 484, 1684, 5661, 18579, 59917, 190696, 600744, 1877256, 5828185, 17998783, 55342617, 169552428, 517884748, 1577812060, 4796682165, 14555626635, 44100374341, 133436026192, 403279293648, 1217616622992, 3673214880049, 11072960931319
OFFSET
0,6
FORMULA
a(n) = A124302(n) - A001519(n).
G.f.: x^4/((x-1)*(3*x-1)*(x^2-3*x+1)).
a(n) = A080936(n,4).
From Colin Barker, Feb 08 2016: (Start)
a(n) = 7*a(n-1)-16*a(n-2)+13*a(n-3)-3*a(n-4) for n>4.
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)
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
EXAMPLE
a(4) = 1 because the only favorable path is UUUUDDDD.
MATHEMATICA
CoefficientList[ Series[x^4/((x-1) (3 x-1) (x^2-3 x+1)), {x, 0, 30}], x].
PROG
(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
(Magma) [((3^(n-1)+1)/2)-Fibonacci(2*n-1): n in [1.. 35]]; // Vincenzo Librandi, Sep 26 2015
(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
CROSSREFS
Column k=4 of A080936.
Sequence in context: A258458 A320546 A066810 * A034577 A372878 A141291
KEYWORD
nonn,easy
AUTHOR
Ran Pan, Sep 25 2015
STATUS
approved