A089503 Triangle of numbers used for basis change between certain falling factorials.

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

Offset

1,3

Comments

Used to relate array A078739 ((2,2)-Stirling2) to triangle A071951 (Legendre-Stirling).

Links

Table of n, a(n) for n=1..42.

Wolfdieter Lang, First 9 rows.

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.
Reformatted - Wolfdieter Lang, Apr 10 2013
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

Sequence in context: A105197 A157398 A306299 * A019236 A019237 A019238

Adjacent sequences: A089500 A089501 A089502 * A089504 A089505 A089506

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Dec 01 2003

Status

approved