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A089505 Triangle of signed numbers used for the computation of the column sequences of triangle A089504.

%I A089505 #13 Aug 28 2019 17:03:45
%S A089505 1,-1,4,1,-24,50,-1,114,-950,1350,31,-15504,400520,-1897200,2052855,
%T A089505 -9269,19612560,-1431859000,17333030000,-56265334125,49236404224,
%U A089505 342953,-3011508588,594221485000,-16634292228125,123422029355625,-302409994743808,222337901418633,-9945637
%N A089505 Triangle of signed numbers used for the computation of the column sequences of triangle A089504.
%C A089505 A089504(n+m,m)= sum(a(m,p)*((p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A089506(m); m=1,2,..., n>=0.
%H A089505 W. Lang, <a href="/A089505/a089505.txt">First 7 rows</a>.
%F A089505 a(n, m)= D(n)*((-1)^(n-m))*(((m+2)*(m+1)*m)^(n-1))/(product(fallfac(m+2, 3)-fallfac(r+2, 3), r=1..m-1)*product(fallfac(r+2, 3)-fallfac(m+2, 3), r=m+1..n)), with D(n) := A089506(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)
%F A089505 a(n, m)= A089506(n)*((-1)^(n-m))*(fallfac(m+2, 3)^(n-1))*(3*m^2+6*m+2)/((n-m)!*(m-1)!*product(fallfac(m+r+2, 2)-r*m, r=1..n)), n>=m>=1.
%e A089505 [1]; [ -1,4]; [1,-24,50]; [ -1,114,-950,1350]; ...
%e A089505 a(3,2)= -24 = 27*(-1)*((4*3*2)^2)/((4*3*2-3*2*1)*(5*4*3-4*3*2)).
%e A089505 A089504(2+3,3) = A089513(2) = 6156 = (1*(3*2*1)^2 - 24*(4*3*2)^2 + 50*(5*4*3)^2)/27.
%t A089505 b[n_, m_] := (-1)^(n - m)*FactorialPower[m + 2, 3]^(n - 1)/(Product[ FactorialPower[m + 2, 3] - FactorialPower[r + 2, 3], {r, 1, m - 1}] * Product[ FactorialPower[r + 2, 3] - FactorialPower[m + 2, 3], {r, m + 1, n}]); den[n_] := LCM @@ Table[ Denominator[b[n, m]], {m, 1, n}]; a[n_, m_] := den[n]*b[n, m]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 02 2016 *)
%Y A089505 Companion denominator sequence is A089506.
%K A089505 sign,easy,tabl
%O A089505 1,3
%A A089505 _Wolfdieter Lang_, Dec 01 2003