A090436 -id:A090436 - cp's OEIS frontend

A090217 A generalization of triangle A071951 (Legendre-Stirling).

1, 120, 1, 14400, 840, 1, 1728000, 619200, 3360, 1, 207360000, 447552000, 9086400, 10080, 1, 24883200000, 322444800000, 23345280000, 76824000, 25200, 1, 2985984000000, 232185139200000, 59152550400000, 539602560000, 457848000

Offset: 1

Wolfdieter Lang, Dec 01 2003

This is the fourth member of the family A071951 (Legendre-Stirling,(2,2) case), A089504((3,3)-case), A090215 ((4,4)-case).

This triangle underlies the array entry A090216 ((5,5)-generalized Stirling2).

			Triangle starts:
[1];
[120,1];
[14400,840,1];
[1728000,619200,3360,1];
...

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 5 rows.

The column sequences (without leading zeros) are powers of 120, etc.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 4, 5]*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 02 2016 *)

G.f. for m-th column (without leading zeros and m>=1) is 1/product(1-fallfac(r+4, 5)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m)=sum(A090435(m, p)*fallfac(p, 5)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A090436(m).

A090435 Triangle of signed numbers used for the computation of the column sequences of triangle A090217.

Original entry on oeis.org

1, -1, 6, 1, -48, 147, -5, 1584, -24255, 50176, 1, -1980, 121275, -1003520, 1571724, -41, 496980, -113458275, 2950635520, -16174611684, 20412000000, 45182, -3322062810, 2744728561050, -206756932157440, 3081396966348393, -12443694076800000, 13160600037440625, -1294492177294

Offset: 1

Wolfdieter Lang, Dec 01 2003

A090217(n+m,m)= sum(a(m,p)*((p+4)*(p+3)*(p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A090436(m); m=1,2,..., n>=0.

			[1]; [ -1,6]; [1,-48,147]; [ -5,1584,-24255,50176]; ...
A090217(2+3,3) = 9086400 = (1*(5*4*3*2*1)^2 - 48*(6*5*4*3*2)^2 + 147*(7*6*5*4*3)^2)/100.
a(3,2)= -48 = 100*(-1)*((6*5*4*3*2)^2)/((6*5*4*3*2-5*4*3*2*1)*(7*6*5*4*3-6*5*4*3*2)).

W. Lang, First 8 rows.

a(n, m)= D(n)*((-1)^(n-m))*(fallfac(m+4, 5)^(n-1))/(product(fallfac(m+4, 5)-fallfac(r+4, 5), r=1..m-1)*product(fallfac(r+4, 5)-fallfac(m+4, 5), r=m+1..n)), with D(n) := A090436(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.)

cp's OEIS Frontend

A090217 A generalization of triangle A071951 (Legendre-Stirling).

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