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

%I #9 Jul 22 2024 15:16:50

%S 1,-1,1,1,-6,1,-1,23,-23,1,1,-76,230,-76,1,-1,237,-1682,1682,-237,1,1,

%T -722,10543,-23548,10543,-722,1,-1,2179,-60657,259723,-259723,60657,

%U -2179,1,1,-6552,331612,-2485288,4675014,-2485288,331612,-6552,1,-1,19673,-1756340,21707972,-69413294,69413294,-21707972,1756340,-19673,1

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

%C 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 G. C. Greubel, <a href="/A138076/b138076.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = (-1)^(n+k) * A060187(n+1, k+1).

%F From _G. C. Greubel_, Jul 21 2024: (Start)

%F T(2*n, n) = (-1)^n * A177043(n).

%F Sum_{k=0..n} T(n, k) = (1/2)*(1 + (-1)^n)*(-1)^floor((n+ 1)/2) * A002436(floor(n/2)).

%F Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A000165(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A178118(n+1). (End)

%e Triangle begins as:

%e 1;

%e -1, 1;

%e 1, -6, 1;

%e -1, 23, -23, 1;

%e 1, -76, 230, -76, 1;

%e -1, 237, -1682, 1682, -237, 1;

%e 1, -722, 10543, -23548, 10543, -722, 1;

%e -1, 2179, -60657, 259723, -259723, 60657, -2179, 1;

%e 1, -6552, 331612, -2485288, 4675014, -2485288, 331612, -6552, 1;

%t p[t_] = Exp[t]*x/(Exp[2*t] + x);

%t Table[CoefficientList[(n!*(1+x)^(n+1)/x)*SeriesCoefficient[Series[p[ t], {t,0,30}], n], x], {n,0,12}]//Flatten

%o (Magma)

%o A060187:= func< n,k | (&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]) >;

%o A138076:= func< n,k | (-1)^(n+k)*A060187(n+1,k+1) >;

%o [A138076(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 21 2024

%o (SageMath)

%o @CachedFunction

%o def t(n,k): # t = A060187

%o if k==1 or k==n: return 1

%o return (2*(n-k)+1)*t(n-1, k-1) + (2*k-1)*t(n-1, k)

%o def A138076(n,k): return (-1)^(n+k)*t(n+1,k+1)

%o flatten([[A138076(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 21 2024

%Y Cf. A000165, A002436, A060187, A177043, A178118.

%K sign,tabl

%O 0,5

%A _Roger L. Bagula_, Nov 26 2009