A089503 - OEIS

%I #31 Nov 17 2021 22:54:43

%S 1,1,4,1,12,30,1,24,168,336,1,40,540,2880,5040,1,60,1320,13200,59400,

%T 95040,1,84,2730,43680,360360,1441440,2162160,1,112,5040,117600,

%U 1528800,11007360,40360320,57657600,1,144,8568,274176,5140800,57576960

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

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

%H Wolfdieter Lang, <a href="/A089503/a089503.txt">First 9 rows</a>.

%F 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.

%F 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

%e The triangle begins:

%e n\m 1   2    3      4       5        6        7        8 ...

%e 1:  1

%e 2:  1   4

%e 3:  1  12   30

%e 4:  1  24  168    336

%e 5:  1  40  540   2880    5040

%e 6:  1  60 1320  13200   59400    95040

%e 7:  1  84 2730  43680  360360  1441440  2162160

%e 8:  1 112 5040 117600 1528800 11007360 40360320 57657600

%e ...

%e Row 9:  1 144 8568 274176 5140800 57576960 374250240 1283143680 1764322560

%e Row 10: 1 180 13680 574560 14651280 234420480 2344204800 14065228800 45711993600 60949324800.

%e Reformatted - _Wolfdieter Lang_, Apr 10 2013

%e n=3: fallfac(x+2,6) = 1*fallfac(x,6) + 12*fallfac(x,5) + 30*fallfac(x,4).

%t 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 *)

%t 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 *)

%K nonn,easy,tabl

%O 1,3

%A _Wolfdieter Lang_, Dec 01 2003