|
|
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.
|
|
3
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
FORMULA
|
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
|
|
|
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]];
|
|
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
if (k==0 or k==n): return 1
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|