login
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, -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
0,5
REFERENCES
Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, McGraw-Hill, New York, 1976, page 91.
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)
EXAMPLE
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;
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 *)
PROG
(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
(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
CROSSREFS
Cf. 033312, A109128.
Sequence in context: A214248 A152719 A107044 * A174545 A102523 A323023
KEYWORD
tabl,sign
AUTHOR
EXTENSIONS
Edited and new name by G. C. Greubel, Sep 15 2024
STATUS
approved