A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

Keywords

• sign
• tabl

Examples

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;
		

Links

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

Crossrefs

Cf. A005809, A045721, A066170.

Row reversal of: A374439.

Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).

Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).

Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

Programs

Magma

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

Mathematica

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

SageMath

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

SageMath

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

Formula

From G. C. Greubel, Jan 22 2025: (Start)

T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).

T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Extensions

Edited by G. C. Greubel, Jan 22 2025