|
|
A177767
|
|
Triangle read by rows: T(n,k) = binomial(n - 1, k - 1), 1 <= k <= n, and T(n,0) = A153881(n+1), n >= 0.
|
|
0
|
|
|
1, -1, 1, -1, 1, 1, -1, 1, 2, 1, -1, 1, 3, 3, 1, -1, 1, 4, 6, 4, 1, -1, 1, 5, 10, 10, 5, 1, -1, 1, 6, 15, 20, 15, 6, 1, -1, 1, 7, 21, 35, 35, 21, 7, 1, -1, 1, 8, 28, 56, 70, 56, 28, 8, 1, -1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, -1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,9
|
|
COMMENTS
|
Row sums yield A000225 preceded by 1.
|
|
LINKS
|
|
|
FORMULA
|
Row n = coefficients in the expansion of x*(1 + x)^(n - 1) - 1, n > 0.
G.f.: (1 - 3*y + (2 + x)*y^2)/(1 - (2 + x)*y + (1 + x)*y^2).
E.g.f.: (2 + x - (1 + x)*exp(y) + x*exp((1 + x)*y))/(1 + x). (End)
|
|
EXAMPLE
|
Triangle begins:
1;
-1, 1;
-1, 1, 1;
-1, 1, 2, 1;
-1, 1, 3, 3, 1;
-1, 1, 4, 6, 4, 1;
-1, 1, 5, 10, 10, 5, 1;
-1, 1, 6, 15, 20, 15, 6, 1;
-1, 1, 7, 21, 35, 35, 21, 7, 1;
-1, 1, 8, 28, 56, 70, 56, 28, 8, 1;
-1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
...
|
|
MATHEMATICA
|
Flatten[Table[If[n == 0, {1}, CoefficientList[x*(1 + x)^( n - 1) - 1, x]], {n, 0, 10}]]
|
|
PROG
|
(Maxima)
T(n, k) := if k = 0 then 2*floor(1/(n + 1)) - 1 else binomial(n - 1, k - 1)$
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|