|
| |
|
|
A081577
|
|
Pascal-(1,2,1) array read by antidiagonals.
|
|
11
| |
|
|
1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 22, 10, 1, 1, 13, 46, 46, 13, 1, 1, 16, 79, 136, 79, 16, 1, 1, 19, 121, 307, 307, 121, 19, 1, 1, 22, 172, 586, 886, 586, 172, 22, 1, 1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1, 1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
COMMENTS
| One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016777, A038764, A081583, A081584. Coefficients of the row polynomials in the Newton basis are given by A013610.
As a number triangle, this is the Riordan array (1/(1-x), x(1+2x)/(1-x)). It has row sums A002605 and diagonal sums A077947. - Paul Barry (pbarry(AT)wit.ie), Jan 24 2005
All entries are == 1 mod 3. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Oct 04 2008
Row sums are A002605. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008)
|
|
|
REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
|
|
|
FORMULA
| Square array T(n, k) defined by T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1)+2T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+2x)^k/(1-x)^(k+1).
G.f.: 1/(1-x-y-2xy). - Ralf Stephan, Apr 28 2004
T(n,k)=sum{j=0..n, C(k,j-k)*C(n+k-j,k)*2^(j-k)}; - Paul Barry (pbarry(AT)wit.ie), Oct 23 2006
a(n)=2*{0,a(n-2),0}+{0,a(n-1)}+{a(n-1),0}. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008]
|
|
|
EXAMPLE
| Rows begin
1 1 1 1 1 ...
1 4 7 10 13 ...
1 7 22 46 79 ...
1 10 46 136 307 ...
1 13 79 307 886 ...
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008: (Start) As a triangle this begins:
{1},
{1, 1},
{1, 4, 1},
{1, 7, 7, 1},
{1, 10, 22, 10, 1},
{1, 13, 46, 46, 13, 1},
{1, 16, 79, 136, 79, 16, 1},
{1, 19, 121, 307, 307, 121, 19, 1},
{1, 22, 172, 586, 886, 586, 172, 22, 1},
{1, 25, 232, 1000, 2086, 2086, 1000, 232, 25, 1},
{1, 28, 301, 1576, 4258, 5944, 4258, 1576, 301, 28, 1} (End)
...
|
|
|
MATHEMATICA
| Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008: (Start)
Clear[a]; a[0] = {1}; a[1] = {1, 1};
a[n_] := a[n] = 2*Join[{0}, a[n - 2], {0}] + Join[{0}, a[n - 1]] + Join[a[n - 1], {0}];
Table[a[n], {n, 0, 10}]; Flatten[%] (End)
|
|
|
CROSSREFS
| Cf. A081578, A081579, A081580.
A008288 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 09 2008]
Sequence in context: A131060 A124376 A047671 * A146986 A146771 A073697
Adjacent sequences: A081574 A081575 A081576 * A081578 A081579 A081580
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 23 2003
|
| |
|
|
