OFFSET
1,1
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).
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
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]
PROG
(PARI) Vec((x^3-3*x^2+3*x-2)/(x-1)^5 + O(x^50)) \\ G. C. Greubel, Dec 21 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
STATUS
approved