A132458 Let df(n,k) = Product_{i=0..k-1} (n-i) be the descending factorial and let P(m,n) = df(n-1,m-1)^2*(2*n-m)/((m-1)!*m!). Sequence gives P(4,n).

0, 0, 0, 1, 24, 200, 1000, 3675, 10976, 28224, 64800, 136125, 266200, 490776, 861224, 1449175, 2352000, 3699200, 5659776, 8450649, 12346200, 17689000, 24901800, 34500851, 47110624, 63480000, 84500000, 111223125, 144884376, 186924024, 239012200, 303075375

Offset: 1

Author

Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007

Keywords

• nonn

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).

For m = 1 through 8 we get respectively A005408, A000578, A108674, this sequence, A133317, A132464, A132465, A132466.

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).

Crossrefs

Cf. A005408, A000578, A108674, A133317, A132464, A132465, A132466.

Programs

Magma

[(n-3)^2*(n-2)^3*(n-1)^2/72: n in [1..40]]; // Vincenzo Librandi, Aug 06 2017

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 *)

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