A177350 Triangle read by rows: T(n,k)=Sum(binomial(n - (m + k), m + k), (m, 1, floor[n/2 + 1]))

-1, 0, 2, 1, -1, 3, 2, 0, 4, 5, 4, 1, -1, 5, 8, 7, 3, 0, 6, 12, 13, 12, 7, 1, -1, 7, 17, 21, 20, 14, 4, 0, 8, 23, 33, 34, 33, 26, 11, 1, -1, 9, 30, 50, 55, 54, 46, 25, 5, 0, 10, 38, 73, 88, 89, 88, 79, 51, 16, 1, -1

Offset

0,3

Comments

Row sums are:{-1, 0, 2, 5, 13, 23, 50, 83, 168, 274, 532,...}.

An effort to generalize the Lucas Fibonacci sum: m limit seems to be off. [Connection with Lucas or Fibonacci sequences is not clear to me, and what does "off" mean? - N. J. A. Sloane, Dec 16 2010]

For comparison: the incomplete generalized Fibonacci Polynomials [arXiv:1308.4192] F_n(k) = sum_{j=0..k} binomial(n-1-j,j) start with a leading column of 1's as:

1;

1;

1 2;

1 3;

1 4 5;

1 5 8;

1 6 12 13;

1 7 17 21;

1 8 23 33 34;

1 9 30 50 55; - R. J. Mathar, Aug 23 2013

Links

Table of n, a(n) for n=0..60.

Formula

k between -Floor[n/2]and Floor[n/2]:

t(n,k)=Sum(Binomial(n - (m + k), m + k), {m, 1, Floor[n/2 + 1]))

Example

-1;
0;
2, 1, -1;
3, 2, 0;
4, 5, 4, 1, -1;
5, 8, 7, 3, 0;
6, 12, 13, 12, 7, 1, -1;
7, 17, 21, 20, 14, 4, 0;
8, 23, 33, 34, 33, 26, 11, 1, -1;
9, 30, 50, 55, 54, 46, 25, 5, 0;
10, 38, 73, 88, 89, 88, 79, 51, 16, 1, -1;

Mathematica

w[n_, m_, k_] = Binomial[n - (m + k), m + k];
t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 + 1]}];
Table[Table[t[n, k], {k, -Floor[n/2], Floor[n/2]}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A000045

Sequence in context: A183568 A291958 A127948 * A190688 A145975 A374683

Adjacent sequences: A177347 A177348 A177349 * A177351 A177352 A177353

Keyword

sign,tabf

Author

Roger L. Bagula, Dec 10 2010

Status

approved