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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A022072 A351221 A154229 * A225398 A037936 A277642

Adjacent sequences: A225430 A225431 A225432 * A225434 A225435 A225436

Keyword

sign,tabl

Author

Roger L. Bagula, May 07 2013

Extensions

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

Edited by G. C. Greubel, Mar 19 2022

Status

approved