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.

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, 2, -1, -1, 2, 114, 604, 604, 114, 2, -1, -1, 2, 240, 2382, 4832, 2382, 240, 2, -1, -1, 2, 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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Aug 20 2008

Keywords

• tabl
• sign

Examples

			Triangle begins as:
   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,      2,     -1;
  -1,  2,  114,   604,    604,    114,      2,    -1;
  -1,  2,  240,  2382,   4832,   2382,    240,     2,   -1;
  -1,  2,  494,  8586,  31238,  31238,   8586,   494,    2,  -1;
  -1,  2, 1004, 29216, 176468, 312380, 176468, 29216, 1004,   2,  -1;
		

References

• Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, McGraw-Hill, New York, 1976, page 91.

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. 033312, A109128.

Programs

Magma

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j)^n: j in [0..k]]) >; // A008292
function A141591(n,k)
  if n eq 0 then return 1;
  elif k eq 0 and n eq 1 then return 1;
  elif k eq 0 or k eq n then return -1;
  else return 2*Eulerian(n-1,k);
  end if;
end function;
[A141591(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 15 2024

Mathematica

(* First program *)
f[x_, n_]:= f[x, n]= (1-x)^(n+1)*Sum[k^n*x^k, {k, 0, Infinity}];
Table[Simplify[f[x, n]], {n, 0, 10}];
Join[{{1}}, Table[Join[CoefficientList[2*f[x,n] -1, x], {-1}], {n, 0, 10}]]//Flatten
(* Second program *)
Eulerian[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}]; (* A008292 *)
A141591[n_, k_]:= If[k==0 || k==n, -1, 2*Eulerian[n-1,k]] +2*Boole[n==0 || n ==1 && k==0];
Table[A141591[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 15 2024 *)

SageMath

@CachedFunction
def A008292(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j)^n for j in range(k+1))
def A141591(n,k):
    if (k==0 and n==0): return 1
    elif (k==0 and n==1): return 1
    elif (k==0 or k==n): return -1
    else: return 2*A008292(n-1, k)
flatten([[A141591(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 15 2024

Formula

T(n, k) = 2*A123125(n-1, k), with T(0, 0) = T(1, 0) = 1, otherwise T(n, 0) = T(n, n) = -1.

Sum_{k=0..n} T(n, k) = 2*033312(n), for n >= 1, otherwise 1 (n=0).

From G. C. Greubel, Sep 15 2024: (Start)

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.

T(n, n-k) = T(n, k) for n >= 2. (End)

Extensions

Edited and new name by G. C. Greubel, Sep 15 2024