A225433 Triangle T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, -38, 1, 1, -165, -165, 1, 1, -676, 4806, -676, 1, 1, -2723, 44452, 44452, -2723, 1, 1, -10914, 362895, -1346780, 362895, -10914, 1, 1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1, 1, -174752, 20554588, -263879264, 683233990, -263879264, 20554588, -174752, 1

Offset: 0

Author

Roger L. Bagula, May 07 2013

Keywords

• sign
• easy
• tabl

Examples

			The triangle begins:
  1;
  1,      1;
  1,    -38,       1;
  1,   -165,    -165,         1;
  1,   -676,    4806,      -676,         1;
  1,  -2723,   44452,     44452,     -2723,       1;
  1, -10914,  362895,  -1346780,    362895,  -10914,      1;
  1, -43681, 2780367, -20297327, -20297327, 2780367, -43681, 1;
		

Links

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

Crossrefs

Cf. A007318, A060187, A142458, A159041, A225356.

Programs

Maple

See Maple program in A159041.

Mathematica

(* First program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n+1) -m*(k+1) +1)*T[n-1, k- 1, m] + (m*(k+1) -(m-1))*T[n-1, k, m] ];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i<= Floor[n/2], (-1)^i*T[n,i,3], -(-1)^(n-i)*T[n,i,3]]], {i,0,n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n,0,12}]]
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A142458[n_, k_]:= T[n,k,3];
A225433[n_, k_]:= A225433[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], A225433[n, k-1] +(-1)^k*A142458[n+2, k+1], A225433[n, n-k]]];
Table[A225433[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 19 2022 *)

Sage

@CachedFunction
def T(n, k, m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m) + (m*k-m+1)*T(n-1, k, m)
def A142458(n,k): return T(n,k,3)
@CachedFunction
def A225433(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225433(n,k-1) + (-1)^k*A142458(n+2,k+1)
    else: return A225433(n,n-k)
flatten([[A225433(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 19 2022

Formula

From G. C. Greubel, Mar 19 2022: (Start)

T(n, k) = T(n, k-1) + (-1)^k*A142458(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

T(n, n-k) = T(n, k). (End)

Extensions

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 19 2022