|
|
A159856
|
|
Triangle read by rows: T(n,0) = n+1, T(n,k) = 2*T(n-1,k) - T(n-1,k-1), T(n,k) = 0 if k > n and if k < 0.
|
|
1
|
|
|
1, 2, -1, 3, -4, 1, 4, -11, 6, -1, 5, -26, 23, -8, 1, 6, -57, 72, -39, 10, -1, 7, -120, 201, -150, 59, -12, 1, 8, -247, 522, -501, 268, -83, 14, -1, 9, -502, 1291, -1524, 1037, -434, 111, -16, 1, 10, -1013, 3084, -4339, 3598, -1905, 656, -143, 18, -1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
A Riordan array - see the Luzon references.
|
|
LINKS
|
|
|
FORMULA
|
Conjecture: Sum_{i=0..n} |T(n,k)| = A047926(n). (End)
T(n,k) = (-1)^k*Sum_{i=0..n-k} binomial(n+1,i+k+1)*binomial(i+k-1,k). - Vladimir Kruchinin, Nov 22 2016
|
|
EXAMPLE
|
Triangle begins
1;
2, -1;
3, -4, 1;
4, -11, 6, -1;
5, -26, 23, -8, 1;
6, -57, 72, -39, 10, -1;
7, -120, 201, -150, 59, -12, 1;
...
|
|
MATHEMATICA
|
With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x)/(1-x)^2/(1-2*x
+y*x), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)
|
|
PROG
|
(Maxima)
T(n, k):=coeff(taylor(1/(1-x)^2*(-x/(1-x))^k, x, 0, 15), x, n); /* Vladimir Kruchinin, Nov 22 2016 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|