|
| |
|
|
A211229
|
|
Matrix inverse of lower triangular array A211226.
|
|
2
|
|
|
|
1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 1, 0, 0, -2, 1, -1, 1, 0, 0, -1, 1, 2, -3, 3, 0, 0, -3, 1, -2, 2, -3, 3, 0, 0, -1, 1, 9, -8, 8, -12, 6, 0, 0, -4, 1, -9, 9, -8, 8, -6, 6, 0, 0, -1, 1, 44, -45, 45, -40, 20, -30, 10, 0, 0, -5, 1, -44, 44, -45, 45, -20, 20, -10, 10, 0, 0, -1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,14
|
|
|
COMMENTS
|
This triangle is related to the derangement numbers. The subtriangles (T(2*n,2*k))n,k>=0, -(T(2*n+1,2*k))n,k>=0, and (T(2*n+1,2*k+1))n,k>=0 are all equal to A008290, while the subtriangle (T(2*n,2*k+1))n,k>=0 equals -A180188 (with an extra initial row of zeros).
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(2*n,2*k) = T(2*n+1,2*k+1) = -T(2*n+1,2*k) = binomial(n,k)*A000166(n-k) = (n!/k!)*Sum_{i = 0..n-k} (-1)^i/i!;
T(2*n,2*k+1) = -n*binomial(n-1,k)*A000166(n-k-1) = -(n!/k!)*Sum_{i = 0..n-k-1} (-1)^i/i!.
Column entries:
With b(j) = floor(j/2); h = 1 for n even and k odd, h = 0 else:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*Sum_{i = 0..b(n-k)-h} (-1)^i/i!.
Sum-free formula:
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!) for n-k < 2.
T(n,k) = (-1)^(n+k)*(b(n)!/b(k)!)*round((b(n-k)-h)!/exp(1))/(b(n-k)-h)!) otherwise. (End)
|
|
|
EXAMPLE
|
Triangle begins:
n\k | 0 1 2 3 4 5 6 7 8 9
=====+==================================================
0 | 1
1 | -1 1
2 | 0 -1 1
3 | 0 0 -1 1
4 | 1 0 0 -2 1
5 | -1 1 0 0 -1 1
6 | 2 -3 3 0 0 -3 1
7 | -2 2 -3 3 0 0 -1 1
8 | 9 -8 8 -12 6 0 0 -4 1
9 | -9 9 -8 8 -6 6 0 0 -1 1
...
|
|
|
MATHEMATICA
|
b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
Table[(-1)^(n+k) (b[n]!/b[k]!) Sum[(-1)^i/i!, {i, 0, b[n-k]-h}], {n, 0, 31}, {k, 0, n}] //Flatten (* Manfred Boergens, Jan 10 2023 *)
(* Sum-free code *)
b[j_] = Floor[j/2]; h = If[EvenQ[n] && OddQ[k], 1, 0];
T[n_, k_] = (-1)^(n+k) (b[n]!/b[k]!) If[n-k<2, 1, Round[(b[n-k]-h)!/E]/(b[n-k]-h)!];
Table[T[n, k], {n, 0, 31}, {k, 0, n}] // Flatten
|
|
|
PROG
|
T(n, k) = f(n)/(f(k)*f(n-k)); \\ A211226
tabl(nn) = my(m=matrix(nn, nn, n, k, if (n>=k, T(n-1, k-1), 0))); 1/m; \\ Michel Marcus, Jan 10 2023
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
