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
Examples
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).
Links
- Wolfdieter Lang, First 9 rows.
Programs
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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 *)
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
Comments