|
|
A115179
|
|
Expansion of c(x*y*(1-x)), c(x) the g.f. of A000108.
|
|
4
|
|
|
1, 0, 1, 0, -1, 2, 0, 0, -4, 5, 0, 0, 2, -15, 14, 0, 0, 0, 15, -56, 42, 0, 0, 0, -5, 84, -210, 132, 0, 0, 0, 0, -56, 420, -792, 429, 0, 0, 0, 0, 14, -420, 1980, -3003, 1430, 0, 0, 0, 0, 0, 210, -2640, 9009, -11440, 4862, 0, 0, 0, 0, 0, -42, 1980, -15015, 40040, -43758, 16796
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
Since C(x*(1-x)) = 1/(1-x), the row sums of this triangle are (1,1,1,...). This establishes the identity Sum_{k=0..n} (-1)^(n-k)*C(k)*binomial(k,n-k)} = 1, where C(x) is the g.f. of A000178.
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = (-1)^(n-k)*binomial(k, n-k)*Catalan(k).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A115178(n) (upward diagonal sums).
T(n, k) = (-1)^(n+k)*A117434(n, k).
|
|
EXAMPLE
|
Triangle begins
1;
0, 1;
0, -1, 2;
0, 0, -4, 5;
0, 0, 2, -15, 14;
0, 0, 0, 15, -56, 42;
0, 0, 0, -5, 84, -210, 132;
0, 0, 0, 0, -56, 420, -792, 429;
|
|
MATHEMATICA
|
Table[(-1)^(n+k)*CatalanNumber[k]*Binomial[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *)
|
|
PROG
|
(Magma) [(-1)^(n+k)*Binomial(k, n-k)*Catalan(k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 31 2021
(Sage) flatten([[(-1)^(n+k)*binomial(k, n-k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|