A268587 Expansion of x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.

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

Offset: 0

Author

Ran Pan, Feb 07 2016

Keywords

• nonn
• easy

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 and 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).

Crossrefs

Cf. A268462, A268586.

Programs

GAP

Concatenation([0,0,0,0], List([3..40], n-> 2^(n-7)*(n-3)*(n+4)*(n+11)/3 )); # G. C. Greubel, May 24 2019

Magma

R:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(5-16*x+13*x^2)/(1-2*x)^4 )); // G. C. Greubel, May 24 2019

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 -16x +13x^2)/(1-2x)^4, {x, 0, 40}], x] (* Michael De Vlieger, Feb 08 2016 *)
LinearRecurrence[{8,-24,32,-16},{0,0,0,0,5,24,85},40] (* Harvey P. Dale, Feb 22 2025 *)

PARI

concat(vector(4), Vec(x^4*(5-16*x+13*x^2)/(1-2*x)^4 + O(x^40))) \\ Colin Barker, Feb 08 2016

Sage

(x^4*(5-16*x+13*x^2)/(1-2*x)^4).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

Formula

G.f.: 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)

E.g.f.: (33 + 60*x + 39*x^2 + (-33 + 6*x + 15*x^2 + 2*x^3)*exp(2*x))/96. - G. C. Greubel, May 24 2019

Extensions

Typo in name and g.f. corrected by Georg Fischer, May 24 2019