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A138076 Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

%I A138076 #9 Jul 22 2024 15:16:50
%S A138076 1,-1,1,1,-6,1,-1,23,-23,1,1,-76,230,-76,1,-1,237,-1682,1682,-237,1,1,
%T A138076 -722,10543,-23548,10543,-722,1,-1,2179,-60657,259723,-259723,60657,
%U A138076 -2179,1,1,-6552,331612,-2485288,4675014,-2485288,331612,-6552,1,-1,19673,-1756340,21707972,-69413294,69413294,-21707972,1756340,-19673,1
%N A138076 Triangle read by rows: T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).
%C A138076 Former name: A signed version of A060187 obtained by taking the Z-transform of p(t,x) = exp(t*(1+2*x)). - _G. C. Greubel_, Jul 21 2024
%H A138076 G. C. Greubel, <a href="/A138076/b138076.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A138076 T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).
%F A138076 From _G. C. Greubel_, Jul 21 2024: (Start)
%F A138076 T(2*n, n) = (-1)^n * A177043(n).
%F A138076 Sum_{k=0..n} T(n, k) = (1/2)*(1 + (-1)^n)*(-1)^floor((n+ 1)/2) * A002436(floor(n/2)).
%F A138076 Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A000165(n).
%F A138076 Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A178118(n+1). (End)
%e A138076 Triangle begins as:
%e A138076    1;
%e A138076   -1,     1;
%e A138076    1,    -6,      1;
%e A138076   -1,    23,    -23,        1;
%e A138076    1,   -76,    230,      -76,       1;
%e A138076   -1,   237,  -1682,     1682,    -237,        1;
%e A138076    1,  -722,  10543,   -23548,   10543,     -722,      1;
%e A138076   -1,  2179, -60657,   259723, -259723,    60657,  -2179,     1;
%e A138076    1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1;
%t A138076 p[t_] = Exp[t]*x/(Exp[2*t] + x);
%t A138076 Table[CoefficientList[(n!*(1+x)^(n+1)/x)*SeriesCoefficient[Series[p[ t], {t,0,30}], n], x], {n,0,12}]//Flatten
%o A138076 (Magma)
%o A138076 A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;
%o A138076 A138076:= func< n,k | (-1)^(n+k)*A060187(n+1,k+1) >;
%o A138076 [A138076(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 21 2024
%o A138076 (SageMath)
%o A138076 @CachedFunction
%o A138076 def t(n,k): # t = A060187
%o A138076     if k==1 or k==n: return 1
%o A138076     return (2*(n-k)+1)*t(n-1, k-1) + (2*k-1)*t(n-1, k)
%o A138076 def A138076(n,k): return (-1)^(n+k)*t(n+1,k+1)
%o A138076 flatten([[A138076(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 21 2024
%Y A138076 Cf. A000165, A002436, A060187, A177043, A178118.
%K A138076 sign,tabl
%O A138076 0,5
%A A138076 _Roger L. Bagula_, Nov 26 2009