|
| |
|
|
A089503
|
|
Triangle of numbers used for basis change between certain falling factorials.
|
|
2
|
|
|
|
1, 1, 4, 1, 12, 30, 1, 24, 168, 336, 1, 40, 540, 2880, 5040, 1, 60, 1320, 13200, 59400, 95040, 1, 84, 2730, 43680, 360360, 1441440, 2162160, 1, 112, 5040, 117600, 1528800, 11007360, 40360320, 57657600, 1, 144, 8568, 274176, 5140800, 57576960
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
Used to relate array A078739 ((2,2)-Stirling2) to triangle A071951 (Legendre-Stirling).
|
|
|
LINKS
|
|
|
|
FORMULA
|
fallfac(x+n-1, 2*n) = Sum_{m=1..n} a(n, m)*fallfac(x, 2*n-(m-1)), n>=1 where fallfac(x, k) := Product_{j=1..k} (x+1-j), with fallfac(n, k) = A068424(n, k) (falling factorials). a(n, m) = 0 if n < m.
T(n, m) = binomial(n-1, m-1)*binomial(2n, m-1)*m!, for 1 <= m <= n, with binomial(n, m) = A007318. - Stefano Negro, Nov 10 2021
|
|
|
EXAMPLE
|
The triangle begins:
n\m 1 2 3 4 5 6 7 8 ...
1: 1
2: 1 4
3: 1 12 30
4: 1 24 168 336
5: 1 40 540 2880 5040
6: 1 60 1320 13200 59400 95040
7: 1 84 2730 43680 360360 1441440 2162160
8: 1 112 5040 117600 1528800 11007360 40360320 57657600
...
Row 9: 1 144 8568 274176 5140800 57576960 374250240 1283143680 1764322560
Row 10: 1 180 13680 574560 14651280 234420480 2344204800 14065228800 45711993600 60949324800.
n=3: fallfac(x+2,6) = 1*fallfac(x,6) + 12*fallfac(x,5) + 30*fallfac(x,4).
|
|
|
MATHEMATICA
|
eq[n_, x_] := Sum[FactorialPower[x, 1 - m + 2*n]*a[n, m], {m, 1, n}] == FactorialPower[x + n - 1, 2*n]; eq[n_] := Table[eq[n, x], {x, n + 1, 2*n}]; row[n_] := First[Table[a[n, m], {m, 1, n}] /. Solve[eq[n]]]; Array[row, 10] // Flatten (* Jean-François Alcover, Sep 02 2016 *)
a[n_, m_]:= Binomial[n-1, m-1]*Binomial[2n, m-1]*Gamma[m]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] (* Stefano Negro, Nov 10 2021 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
