A110665 Sequence is {a(0,n)}, where a(m,0)=0, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n) is such that a(n,n) = n for all n.

0, 1, 0, -3, -4, 0, 6, 7, 0, -9, -10, 0, 12, 13, 0, -15, -16, 0, 18, 19, 0, -21, -22, 0, 24, 25, 0, -27, -28, 0, 30, 31, 0, -33, -34, 0, 36, 37, 0, -39, -40, 0, 42, 43, 0, -45, -46, 0, 48, 49, 0, -51, -52, 0, 54, 55, 0, -57, -58, 0, 60, 61, 0, -63, -64, 0, 66, 67, 0, -69, -70, 0, 72, 73, 0, -75, -76, 0, 78, 79

Offset: 0

Author

Leroy Quet, Aug 02 2005

Keywords

• easy
• sign

Examples

			a(0,n): 0, 1, 0, -3, -4,...
a(1,n): 0, 1, 1, -2, -6,...
a(2,n): 0, 1, 2, 0, -6,...
a(3,n): 0, 1, 3, 3, -3,...
a(4,n): 0, 1, 4, 7, 4,...
Main diagonal of array is 0, 1, 2, 3, 4,...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Crossrefs

Cf. A110666 - A110672.

Programs

Mathematica

LinearRecurrence[{2,-3,2,-1},{0,1,0,-3},80] (* Harvey P. Dale, Dec 19 2015 *)
a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
Table[a[0, n], {n, 0, 79}] (* Jean-François Alcover, Mar 29 2020 *)

PARI

my(x='x+O('x^50)); concat([0], Vec((x-2*x^2)/(1-x+x^2)^2)) \\ G. C. Greubel, Sep 03 2017

Formula

a(0, n) = n - Sum_{k=0..(n-1)} binomial(2*n-k-1, n-1)*a(0, k).

From Franklin T. Adams-Watters, May 12 2006: (Start)

a(n) = n * A010892(n), where A010892 is periodic sequence [1,1,0,-1,-1,0].

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

Extensions

More terms from Franklin T. Adams-Watters, May 12 2006