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

%I A089503 #31 Nov 17 2021 22:54:43
%S A089503 1,1,4,1,12,30,1,24,168,336,1,40,540,2880,5040,1,60,1320,13200,59400,
%T A089503 95040,1,84,2730,43680,360360,1441440,2162160,1,112,5040,117600,
%U A089503 1528800,11007360,40360320,57657600,1,144,8568,274176,5140800,57576960
%N A089503 Triangle of numbers used for basis change between certain falling factorials.
%C A089503 Used to relate array A078739 ((2,2)-Stirling2) to triangle A071951 (Legendre-Stirling).
%H A089503 Wolfdieter Lang, <a href="/A089503/a089503.txt">First 9 rows</a>.
%F A089503 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 A089503 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 A089503 The triangle begins:
%e A089503 n\m 1   2    3      4       5        6        7        8 ...
%e A089503 1:  1
%e A089503 2:  1   4
%e A089503 3:  1  12   30
%e A089503 4:  1  24  168    336
%e A089503 5:  1  40  540   2880    5040
%e A089503 6:  1  60 1320  13200   59400    95040
%e A089503 7:  1  84 2730  43680  360360  1441440  2162160
%e A089503 8:  1 112 5040 117600 1528800 11007360 40360320 57657600
%e A089503 ...
%e A089503 Row 9:  1 144 8568 274176 5140800 57576960 374250240 1283143680 1764322560
%e A089503 Row 10: 1 180 13680 574560 14651280 234420480 2344204800 14065228800 45711993600 60949324800.
%e A089503 Reformatted - _Wolfdieter Lang_, Apr 10 2013
%e A089503 n=3: fallfac(x+2,6) = 1*fallfac(x,6) + 12*fallfac(x,5) + 30*fallfac(x,4).
%t A089503 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 A089503 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 A089503 nonn,easy,tabl
%O A089503 1,3
%A A089503 _Wolfdieter Lang_, Dec 01 2003