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A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.

%I A225483 #17 Mar 31 2022 10:17:43
%S A225483 1,1,-26,1,1,-120,1192,-120,1,1,-502,14609,-88736,14609,-502,1,1,
%T A225483 -2036,152638,-2205524,9890752,-2205524,152638,-2036,1,1,-8178,
%U A225483 1479727,-45541628,424761262,-1551163136,424761262,-45541628,1479727,-8178,1
%N A225483 Triangle T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j), read by rows.
%H A225483 G. C. Greubel, <a href="/A225483/b225483.txt">Rows n = 0..50 if the irregular triangle, flattened</a>
%F A225483 T(n, k) = [x^k]( A159041(x,n)/(x+1) ).
%F A225483 From _G. C. Greubel_, Mar 29 2022: (Start)
%F A225483 T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A159041(2*n+1, j).
%F A225483 T(n, 2*n-k) = T(n, k). (End)
%e A225483 The triangle begins:
%e A225483   1;
%e A225483   1,   -26,      1;
%e A225483   1,  -120,   1192,     -120,       1;
%e A225483   1,  -502,  14609,   -88736,   14609,     -502,      1;
%e A225483   1, -2036, 152638, -2205524, 9890752, -2205524, 152638, -2036, 1;
%t A225483 (* First program *)
%t A225483 Needs["Combinatorica`"];
%t A225483 p[n_, x_]:= p[n,x]= Sum[If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<=Floor[n/2], (-1)^i*Eulerian[n+1, i]*x^i, (-1)^(n-i+1)*Eulerian[n+1, i]*x^i]], {i,0,n}]/(1- x^2);
%t A225483 Table[CoefficientList[p[x, 2*n], x], {n,0,10}]//Flatten
%t A225483 (* Second program *)
%t A225483 A008292[n_, k_]:= A008292[n, k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
%t A225483 f[n_, k_]:= f[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], f[n, k-1] + (-1)^k*A008292[n+2, k+1], f[n, n-k]]]; (* f = A159041 *)
%t A225483 T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*f[2*n+1,j], {j,0,k}];
%t A225483 Table[T[n, k], {n,0,10}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Mar 29 2022 *)
%o A225483 (Sage)
%o A225483 def A008292(n, k): return sum( (-1)^j*(k-j)^n*binomial(n+1, j) for j in (0..k) )
%o A225483 @CachedFunction
%o A225483 def f(n, k): # A159041
%o A225483     if (k==0 or k==n): return 1
%o A225483     elif (k <= (n//2)): return f(n, k-1) + (-1)^k*A008292(n+2, k+1)
%o A225483     else: return f(n, n-k)
%o A225483 def A225483(n,k): return sum( (-1)^(k-j)*f(2*n+1,j) for j in (0..k) )
%o A225483 flatten([[A225483(n, k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 29 2022
%Y A225483 Cf. A008292, A060187, A158041.
%K A225483 sign,tabf
%O A225483 0,3
%A A225483 _Roger L. Bagula_, May 08 2013
%E A225483 Edited by _G. C. Greubel_, Mar 29 2022