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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (7,-16,13,-3). 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 Cf. A001519, A124302. Column k=4 of A080936. Sequence in context: A258458 A320546 A066810 * A034577 A372878 A141291 Adjacent sequences: A262597 A262598 A262599 * A262601 A262602 A262603 KEYWORD nonn,easy AUTHOR Ran Pan, Sep 25 2015 STATUS approved

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Last modified September 10 06:17 EDT 2024. Contains 375773 sequences. (Running on oeis4.)