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A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504.

%I A090215 #31 Feb 08 2023 10:25:05
%S A090215 1,24,1,576,144,1,13824,17856,504,1,331776,2156544,199296,1344,1,
%T A090215 7962624,259117056,73903104,1328256,3024,1,191102976,31102009344,
%U A090215 26864234496,1189638144,6408576,6048,1,4586471424,3732432224256,9702226427904,1026160275456,11956045824,24697728,11088,1
%N A090215 A generalization of triangles A071951 (Legendre-Stirling) and A089504.
%C A090215 This triangle underlies the array entry A090214 ((4,4)-generalized Stirling2).
%H A090215 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 A090215 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 A090215 Wolfdieter Lang, <a href="/A090215/a090215.txt">First 8 rows</a>.
%F A090215 G.f. for m-th column sequence (without leading zeros and m>=1) is 1/product(1-fallfac(r+3, 4)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).
%F A090215 a(n, m) = sum(A089515(m, p)*fallfac(p, 4)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A089516(m).
%e A090215 [1]; [24,1]; [576,144,1]; [13824,17856,504,1]; ...
%t A090215 max = 10; f[m_] := 1/Product[1-FactorialPower[r+3, 4]*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 A090215 Cf. A071951 (Legendre-Stirling, (2, 2) case), A089504 ((3, 3)-case).
%Y A090215 The column sequences (without leading zeros) are A009968 (powers of 24), etc.
%K A090215 nonn,easy,tabl
%O A090215 1,2
%A A090215 _Wolfdieter Lang_, Dec 01 2003
%E A090215 More terms coming from a-file added by _Michel Marcus_, Feb 08 2023