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A039763 Triangle of D-analogs of Stirling numbers of first kind, rows reversed.

%I A039763 #33 Feb 03 2025 12:08:11
%S A039763 1,1,0,1,-2,1,1,-6,11,-6,1,-12,50,-84,45,1,-20,150,-520,809,-420,1,
%T A039763 -30,355,-2100,6439,-9390,4725,1,-42,721,-6510,33019,-92358,127539,
%U A039763 -62370,1,-56,1316,-16856,127694,-578984,1505524,-1984584,945945,1,-72,2220,-38304,405174,-2702448,11228300,-27491616,34812945,-16216200
%N A039763 Triangle of D-analogs of Stirling numbers of first kind, rows reversed.
%C A039763 |T(n,k)|, 0 <= k <= n, is the number of elements in the Coxeter group D_n with absolute length k. - _Jose Bastidas_, Jul 16 2023
%H A039763 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 A039763 From _Petros Hadjicostas_, Jul 11 2020: (Start)
%F A039763 T(n,k) = A039762(n,n-k) for k = 0..n.
%F A039763 T(n,0) = 1 for n >= 0.
%F A039763 T(n,n) = (-1)^n*(n-1)*(2*n-3)!! for n >= 2.
%F A039763 T(n,k) = [x^(n-k)] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 and k = 0..n. (End)
%e A039763 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A039763   1;
%e A039763   1,   0;
%e A039763   1,  -2,   1;
%e A039763   1,  -6,  11,   -6;
%e A039763   1, -12,  50,  -84,  45;
%e A039763   1, -20, 150, -520, 809, -420;
%e A039763   ...
%o A039763 (PARI) row(n) = if(n==0, [1], Vec(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ _Petros Hadjicostas_, Jul 12 2020
%Y A039763 Cf. A039762 (transposed triangle).
%K A039763 tabl,sign
%O A039763 0,5
%A A039763 Ruedi Suter (suter(AT)math.ethz.ch)
%E A039763 More terms from _Petros Hadjicostas_, Jul 12 2020