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%I #30 Aug 28 2019 16:45:25
%S 1,6,1,36,30,1,216,756,90,1,1296,18360,6156,210,1,7776,441936,387720,
%T 31356,420,1,46656,10614240,23705136,4150440,119556,756,1,279936,
%U 254788416,1432922400,521757936,29257200,373572,1260,1,1679616
%N A generalization of triangle A071951 (Legendre-Stirling).
%C This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).
%C For the computation of the column sequences see A089505.
%H 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 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 W. Lang, <a href="/A089504/a089504.txt">First 8 rows</a>.
%F 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).
%F 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).
%e [1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
%e a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.
%t 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 *)
%Y Cf. A071951 (Legendre-Stirling, (2, 2) case).
%Y The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
%Y Cf. A008279, A089505, A089506.
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_, Dec 01 2003