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A090217
A generalization of triangle A071951 (Legendre-Stirling).
4
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
1,2
COMMENTS
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).
LINKS
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.
FORMULA
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).
EXAMPLE
Triangle starts:
[1];
[120,1];
[14400,840,1];
[1728000,619200,3360,1];
...
MATHEMATICA
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 *)
CROSSREFS
The column sequences (without leading zeros) are powers of 120, etc.
Sequence in context: A267428 A156739 A326798 * A267570 A267286 A062829
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Dec 01 2003
STATUS
approved