OFFSET
1,5
COMMENTS
P(m,n) is the number of n-step paths that start from (0,0) and reach (m,m) for the first time, where the steps are of the following 4 types: N=(x,y)->(x,y+1), E=(x,y)->(x+1,y), NE=(x,y)->(x+1,y+1), LOOP=(x,y)->(x,y).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
FORMULA
From Bruno Berselli, Dec 29 2010: (Start)
a(n) = (n-3)^2*(n-2)^3*(n-1)^2/72.
G.f.: x^4*(1+16*x+36*x^2+16*x^3+x^4)/(1-x)^8. (End)
Sum_{n>=4} 1/a(n) = 72*zeta(3) - 171/2. - Jaume Oliver Lafont, Aug 06 2017
Sum_{n>=4} (-1)^n/a(n) = 531/2 - 288*log(2) - 54*zeta(3). - Amiram Eldar, Sep 20 2022
MAPLE
df:=proc(n, k) mul(n-i, i=0..k-1); end; P:=proc(n, k) df(k-1, n-1)^2*(2*k-n)/((n-1)!*n!); end; [seq(P(4, n), n=1..50)];
MATHEMATICA
CoefficientList[Series[x^3 (1 + 16 x + 36 x^2 + 16 x^3 + x^4) / (1 - x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Aug 06 2017 *)
PROG
(Magma) [(n-3)^2*(n-2)^3*(n-1)^2/72: n in [1..40]]; // Vincenzo Librandi, Aug 06 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007
STATUS
approved