|
| |
|
|
A111006
|
|
Another version of Fibonacci-Pascal triangle A037027.
|
|
3
| |
|
|
1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 5, 5, 0, 0, 0, 3, 10, 8, 0, 0, 0, 1, 9, 20, 13, 0, 0, 0, 0, 4, 22, 38, 21, 0, 0, 0, 0, 1, 14, 51, 71, 34, 0, 0, 0, 0, 0, 5, 40, 111, 130, 55, 0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89, 0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,6
|
|
|
COMMENTS
| Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Row sums are the Jacobsthal numbers A001045(n+1) and column sums form Pell numbers A000129.
Maximal column entries: A038149 = {1, 1, 2, 5, 10, 22, ...}.
T(n,k) gives a convolved Fibonacci sequence (A001629, A001872, ...).
|
|
|
FORMULA
| T(0, 0) = 1, T(n, k) = 0 for k<0 or for n<k, T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).
T(n, k) = A037027(k, n-k) . T(n, n) = A000045(n+1) . T(3n, 2n) = (n+1)*A001002(n+1) = A038112(n).
G.f.: 1/(1-yx(1-x)-x^2*y^2); - Paul Barry (pbarry(AT)wit.ie), Oct 04 2005
Sum_{0, 0<=k<=n} x^k*T(n,k)=(-1)^n*A053524(n+1), (-1)^n*A083858(n+1), (-1)^n*A002605(n), A033999(n), A000007(n), A001045(n+1), A083099(n)for x= -4, -3, -2, -1, 0, 1, 2 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2006
|
|
|
EXAMPLE
| Triangle begins:
.1;
.0, 1;
.0, 1, 2;
.0, 0, 2, 3;
.0, 0, 1, 5, 5;
.0, 0, 0, 3, 10, 8;
.0, 0, 0, 1, 9, 20, 13;
.0, 0, 0, 0, 4, 22, 38, 21;
.0, 0, 0, 0, 1, 14, 51, 71, 34;
.0, 0, 0, 0, 0, 5, 40, 111, 130, 55;
.0, 0, 0, 0, 0, 1, 20, 105, 233, 235, 89;
.0, 0, 0, 0, 0, 0, 6, 65, 256, 474, 420, 144;
|
|
|
CROSSREFS
| Cf. A000045, A000129, A001045, A037027, A038112, A038149, A084938.
Sequence in context: A128541 A122908 A091008 * A046742 A174739 A203994
Adjacent sequences: A111003 A111004 A111005 * A111007 A111008 A111009
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 02 2005
|
| |
|
|
