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A142073 Irregular triangle, T(n, k) = coefficients of p(x, n), where p(x, n) = (1-2*x)^(n+1) * Sum_{j>=0} j^n*(x/(1-x))^j, read by rows.

%I A142073 #17 May 27 2024 07:16:19
%S A142073 1,1,-1,1,-1,1,1,-4,2,1,7,-16,8,1,21,-28,-26,48,-16,1,51,32,-356,408,
%T A142073 -136,1,113,492,-1774,1072,912,-1088,272,1,239,2592,-5008,-6656,20736,
%U A142073 -15872,3968,1,493,10628,-50,-94432,154528,-57856,-45056,39680,-7936,1,1003,38768,108820,-621352,455608,848384,-1538816,884480,-176896
%N A142073 Irregular triangle, T(n, k) = coefficients of p(x, n), where p(x, n) = (1-2*x)^(n+1) * Sum_{j>=0} j^n*(x/(1-x))^j, read by rows.
%C A142073 Except for n=0, the row sums are zero.
%H A142073 G. C. Greubel, <a href="/A142073/b142073.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A142073 T(n, k) = [x^k]( p(x, n) ), where p(x, n) = (1-2*x)^(n+1) * Sum_{j>=0} j^n*(x/(1-x))^j, or  p(x, n) = (1-2*x)^(n+1)*PolyLog(-n, x/(1-x))/x.
%F A142073 T(n, k) = [x^k]( f(x, n) ), where f(x, n) = Sum_{j=0..n} Eulerian(n, j)*(x-1)^j. - Mourad Rahmani (mrahmani(AT)usthb.dz), Aug 29 2010
%e A142073 Irregular triangle begins as:
%e A142073   1;
%e A142073   1,  -1;
%e A142073   1,  -1;
%e A142073   1,   1,    -4,     2;
%e A142073   1,   7,   -16,     8;
%e A142073   1,  21,   -28,   -26,     48,    -16;
%e A142073   1,  51,    32,  -356,    408,   -136;
%e A142073   1, 113,   492, -1774,   1072,    912,  -1088,    272;
%e A142073   1, 239,  2592, -5008,  -6656,  20736, -15872,   3968;
%e A142073   1, 493, 10628,   -50, -94432, 154528, -57856, -45056, 39680, -7936;
%t A142073 p[x_, n_]= If[n==0, 1, (1-2*x)^(n+1)*Sum[k^n*(x/(1-x))^k, {k,0, Infinity}]/x];
%t A142073 Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
%o A142073 (Magma)
%o A142073 m:=12;
%o A142073 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A142073 b:= func< n | n eq 0 select 0 else 2*Floor((n+1)/2) -1 >;
%o A142073 Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j)^n: j in [0..k]]) >;
%o A142073 p:= func< n,x | (&+[Eulerian(n,j)*(x-1)^j: j in [0..n]]) >;
%o A142073 T:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
%o A142073 [T(n,n-k): k in [0..b(n)], n in [0..m]]; // _G. C. Greubel_, May 26 2024
%o A142073 (SageMath)
%o A142073 m=12
%o A142073 def b(n): return 2*int((n+1)/2) - 1 + int(n==0)
%o A142073 def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in range(k+1))
%o A142073 def p(x,n): return sum(Eulerian(n,j)*(x-1)^j for j in range(n+1))
%o A142073 def T(n,k): return ( p(x,n) ).series(x, n+1).list()[k]
%o A142073 flatten([[T(n,n-k) for k in range(b(n)+1)] for n in range(m+1)]) # _G. C. Greubel_, May 26 2024
%Y A142073 Cf. A008292, A141720.
%K A142073 sign,tabf
%O A142073 0,8
%A A142073 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008
%E A142073 Edited by _G. C. Greubel_, May 26 2024