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A141591 Triangle, read by rows, T(n, k) = 2*A123125(n-1, k), for n >= 2, otherwise T(n, 0) = T(n, n) = -1, with T(0, 0) = T(1, 0) = 1.

%I A141591 #12 Sep 18 2024 11:23:57
%S A141591 1,1,-1,-1,2,-1,-1,2,2,-1,-1,2,8,2,-1,-1,2,22,22,2,-1,-1,2,52,132,52,
%T A141591 2,-1,-1,2,114,604,604,114,2,-1,-1,2,240,2382,4832,2382,240,2,-1,-1,2,
%U A141591 494,8586,31238,31238,8586,494,2,-1,-1,2,1004,29216,176468,312380,176468,29216,1004,2,-1,-1,2,2026,95680,910384,2620708,2620708,910384,95680,2026,2,-1
%N A141591 Triangle, read by rows, T(n, k) = 2*A123125(n-1, k), for n >= 2, otherwise T(n, 0) = T(n, n) = -1, with T(0, 0) = T(1, 0) = 1.
%D A141591 Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, McGraw-Hill, New York, 1976, page 91.
%H A141591 G. C. Greubel, <a href="/A141591/b141591.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A141591 T(n, k) = 2*A123125(n-1, k), with T(0, 0) = T(1, 0) = 1, otherwise T(n, 0) = T(n, n) = -1.
%F A141591 Sum_{k=0..n} T(n, k) = 2*033312(n), for n >= 1, otherwise 1 (n=0).
%F A141591 From _G. C. Greubel_, Sep 15 2024: (Start)
%F A141591 T(n, k) = 2*A008292(n, k) for n >= 2, 1 <= k <= n-1, with T(n, 0) = T(n, n) = -1, T(0, 0) = T(1, 0) = 1.
%F A141591 T(n, n-k) = T(n, k) for n >= 2. (End)
%e A141591 Triangle begins as:
%e A141591    1;
%e A141591    1, -1;
%e A141591   -1,  2,   -1;
%e A141591   -1,  2,    2,    -1;
%e A141591   -1,  2,    8,     2,     -1;
%e A141591   -1,  2,   22,    22,      2,     -1;
%e A141591   -1,  2,   52,   132,     52,      2,     -1;
%e A141591   -1,  2,  114,   604,    604,    114,      2,    -1;
%e A141591   -1,  2,  240,  2382,   4832,   2382,    240,     2,   -1;
%e A141591   -1,  2,  494,  8586,  31238,  31238,   8586,   494,    2,  -1;
%e A141591   -1,  2, 1004, 29216, 176468, 312380, 176468, 29216, 1004,   2,  -1;
%t A141591 (* First program *)
%t A141591 f[x_, n_]:= f[x, n]= (1-x)^(n+1)*Sum[k^n*x^k, {k, 0, Infinity}];
%t A141591 Table[Simplify[f[x, n]], {n, 0, 10}];
%t A141591 Join[{{1}}, Table[Join[CoefficientList[2*f[x,n] -1, x], {-1}], {n, 0, 10}]]//Flatten
%t A141591 (* Second program *)
%t A141591 Eulerian[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}]; (* A008292 *)
%t A141591 A141591[n_, k_]:= If[k==0 || k==n, -1, 2*Eulerian[n-1,k]] +2*Boole[n==0 || n ==1 && k==0];
%t A141591 Table[A141591[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 15 2024 *)
%o A141591 (Magma)
%o A141591 Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j)^n: j in [0..k]]) >; // A008292
%o A141591 function A141591(n,k)
%o A141591   if n eq 0 then return 1;
%o A141591   elif k eq 0 and n eq 1 then return 1;
%o A141591   elif k eq 0 or k eq n then return -1;
%o A141591   else return 2*Eulerian(n-1,k);
%o A141591   end if;
%o A141591 end function;
%o A141591 [A141591(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 15 2024
%o A141591 (SageMath)
%o A141591 @CachedFunction
%o A141591 def A008292(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j)^n for j in range(k+1))
%o A141591 def A141591(n,k):
%o A141591     if (k==0 and n==0): return 1
%o A141591     elif (k==0 and n==1): return 1
%o A141591     elif (k==0 or k==n): return -1
%o A141591     else: return 2*A008292(n-1, k)
%o A141591 flatten([[A141591(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 15 2024
%Y A141591 Cf. 033312, A109128.
%K A141591 tabl,sign
%O A141591 0,5
%A A141591 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 20 2008
%E A141591 Edited and new name by _G. C. Greubel_, Sep 15 2024