A089506 -id:A089506 - cp's OEIS frontend

A089504 A generalization of triangle A071951 (Legendre-Stirling).

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

			[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case).
The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
Cf. A008279, A089505, A089506.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/Product_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

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

Original entry on oeis.org

1, -1, 4, 1, -24, 50, -1, 114, -950, 1350, 31, -15504, 400520, -1897200, 2052855, -9269, 19612560, -1431859000, 17333030000, -56265334125, 49236404224, 342953, -3011508588, 594221485000, -16634292228125, 123422029355625, -302409994743808, 222337901418633, -9945637

Offset: 1

Wolfdieter Lang, Dec 01 2003

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.

			[1]; [ -1,4]; [1,-24,50]; [ -1,114,-950,1350]; ...
a(3,2)= -24 = 27*(-1)*((4*3*2)^2)/((4*3*2-3*2*1)*(5*4*3-4*3*2)).
A089504(2+3,3) = A089513(2) = 6156 = (1*(3*2*1)^2 - 24*(4*3*2)^2 + 50*(5*4*3)^2)/27.

W. Lang, First 7 rows.

Companion denominator sequence is A089506.

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 *)

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.)

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.

cp's OEIS Frontend

A089504 A generalization of triangle A071951 (Legendre-Stirling).

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