A051743 a(n) = (1/24)*n*(n + 5)*(n^2 + n + 6).

2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, 1377, 1833, 2394, 3075, 3892, 4862, 6003, 7334, 8875, 10647, 12672, 14973, 17574, 20500, 23777, 27432, 31493, 35989, 40950, 46407, 52392, 58938, 66079, 73850, 82287, 91427, 101308, 111969, 123450

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999

Keywords

• nonn
• easy

Comments

This is exactly the number of directed column-convex polyominoes. [Something is clearly missing from this sentence; as it stands, it makes no reference to the index n. - Jon E. Schoenfield, Dec 20 2016]

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n-3)=coeff(charpoly(A,x),x^(n-4)). [Milan Janjic, Jan 24 2010]

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. A005435, A006027, A105450.

Programs

Mathematica

Table[(n (n + 5) (n^2 + n + 6))/24, {n, 50}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {2, 7, 18, 39, 75}, 50]

PARI

Vec((x^3-3*x^2+3*x-2)/(x-1)^5 + O(x^50)) \\ G. C. Greubel, Dec 21 2016

Formula

a(n) = binomial(n+3, n-1) + binomial(n, n-1) = binomial(n+3, 4) + binomial(n, 1), n > 0.

From Harvey P. Dale, Nov 29 2011: (Start)

a(1)=2, a(2)=7, a(3)=18, a(4)=39, a(5)=75, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: (x^3-3*x^2+3*x-2)/(x-1)^5. (End)

E.g.f.: (1/24)*(48*x + 36*x^2 + 12*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 21 2016