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A268587 Expansion of 2*x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4. 5
0, 0, 0, 0, 5, 24, 85, 264, 760, 2080, 5488, 14080, 35328, 87040, 211200, 505856, 1198080, 2809856, 6533120, 15073280, 34537472, 78643200, 178061312, 401080320, 899153920, 2006974464, 4461690880, 9881780224, 21810380800, 47982837760, 105243475968 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly three east steps below y = x - 1 and no east steps above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.

LINKS

Robert Israel, Table of n, a(n) for n = 0..3270

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

FORMULA

G.f.: 2*x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.

From Colin Barker, Feb 08 2016: (Start)

a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-3)-16*a(n-4) for n>6.

a(n) = 2^(n-7)*(n-3)*(n+4)*(n+11)/3 for n>2.

(End)

MAPLE

F:= gfun:-rectoproc({16*a(n)-32*a(n+1)+24*a(n+2)-8*a(n+3)+a(n+4), a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=5, a(5)=24, a(6)=85}, a(n), remember):

map(F, [$0..40]); # Robert Israel, Feb 07 2016

MATHEMATICA

CoefficientList[Series[x^4 (5 - 16 x + 13 x^2)/(1 - 2 x)^4, {x, 0, 30}], x] (* Michael De Vlieger, Feb 08 2016 *)

PROG

(PARI) concat(vector(4), Vec(x^4*(5-16*x+13*x^2)/(1-2*x)^4 + O(x^100))) \\ Colin Barker, Feb 08 2016

CROSSREFS

Cf. A268462, A268586.

Sequence in context: A213766 A000347 A270906 * A270682 A272420 A089095

Adjacent sequences:  A268584 A268585 A268586 * A268588 A268589 A268590

KEYWORD

nonn,easy

AUTHOR

Ran Pan, Feb 07 2016

STATUS

approved

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.