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A039762 Triangle of D-analogs of Stirling numbers of first kind.

%I A039762 #39 Jul 12 2020 14:55:06
%S A039762 1,0,1,1,-2,1,-6,11,-6,1,45,-84,50,-12,1,-420,809,-520,150,-20,1,4725,
%T A039762 -9390,6439,-2100,355,-30,1,-62370,127539,-92358,33019,-6510,721,-42,
%U A039762 1,945945,-1984584,1505524,-578984,127694,-16856,1316,-56,1,-16216200,34812945,-27491616,11228300,-2702448,405174,-38304,2220,-72,1
%N A039762 Triangle of D-analogs of Stirling numbers of first kind.
%H A039762 Ruedi Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
%F A039762 From _Petros Hadjicostas_, Jul 11 2020: (Start)
%F A039762 T(n,k) = [x^k] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 with T(0,0) = 1. (Empty products equal 1.)
%F A039762 Let R(n,k) = A039757(n,k) = A039758(n,n-k). Then, for n >= 1:
%F A039762 T(n,0) = -(n - 1)*R(n-1,0);
%F A039762 T(n,k) = R(n-1,k-1) - (n - 1)*R(n-1,k) for k = 1..n-1;
%F A039762 T(n,n) = R(n-1, n-1) = 1.
%F A039762 As a result, for n >= 2, T(n,0) = (-1)^n*(n-1)*(2*n-3)!!. (End)
%e A039762 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A039762      1;
%e A039762      0,   1;
%e A039762      1,  -2,    1;
%e A039762     -6,  11,   -6,   1;
%e A039762     45, -84,   50, -12,   1;
%e A039762   -420, 809, -520, 150, -20, 1;
%e A039762   ...
%o A039762 (PARI) row(n) = if(n==0, [1], Vecrev(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ _Petros Hadjicostas_, Jul 12 2020
%Y A039762 Cf. A039757, A039758, A039763 (transposed triangle).
%K A039762 tabl,sign,easy,nice
%O A039762 0,5
%A A039762 Ruedi Suter (suter(AT)math.ethz.ch)
%E A039762 More terms from _Petros Hadjicostas_, Jul 12 2020