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

%I A090217 #32 Aug 29 2019 16:03:02
%S A090217 1,120,1,14400,840,1,1728000,619200,3360,1,207360000,447552000,
%T A090217 9086400,10080,1,24883200000,322444800000,23345280000,76824000,25200,
%U A090217 1,2985984000000,232185139200000,59152550400000,539602560000,457848000
%N A090217 A generalization of triangle A071951 (Legendre-Stirling).
%C A090217 This is the fourth member of the family A071951 (Legendre-Stirling,(2,2) case), A089504((3,3)-case), A090215 ((4,4)-case).
%C A090217 This triangle underlies the array entry A090216 ((5,5)-generalized Stirling2).
%H A090217 R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, <a href="http://arxiv.org/abs/1302.4694">Dually weighted Stirling-type sequences</a>, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
%H A090217 R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, <a href="http://dx.doi.org/10.1016/j.ejc.2014.06.010">Dually weighted Stirling-type sequences</a>, Europ. J. Combin., 43, 2015, 55-67.
%H A090217 W. Lang, <a href="/A090217/a090217.txt">First 5 rows</a>.
%F A090217 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).
%F A090217 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).
%e A090217 Triangle starts:
%e A090217 [1];
%e A090217 [120,1];
%e A090217 [14400,840,1];
%e A090217 [1728000,619200,3360,1];
%e A090217 ...
%t A090217 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 *)
%Y A090217 The column sequences (without leading zeros) are powers of 120, etc.
%K A090217 nonn,easy,tabl
%O A090217 1,2
%A A090217 _Wolfdieter Lang_, Dec 01 2003