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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)

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: A229515 A258458 A066810 * A034577 A141291 A278027

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 26 05:09 EDT 2017. Contains 292502 sequences.