|
|
A225398
|
|
Triangle read by rows: absolute values of odd-numbered rows of A225433.
|
|
4
|
|
|
1, 1, 38, 1, 1, 676, 4806, 676, 1, 1, 10914, 362895, 1346780, 362895, 10914, 1, 1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1, 1, 2796190, 1063096365, 35677598760, 267248150610, 554291429748, 267248150610, 35677598760, 1063096365, 2796190, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = Sum_{j=0..k-1} (-1)^(k-j-1)*A142458(2*n, j+1).
T(n, n-k) = T(n, k). (End)
|
|
EXAMPLE
|
Triangle begins:
1;
1, 38, 1;
1, 676, 4806, 676, 1;
1, 10914, 362895, 1346780, 362895, 10914, 1;
1, 174752, 20554588, 263879264, 683233990, 263879264, 20554588, 174752, 1;
|
|
MATHEMATICA
|
(* First 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];
T[n_, k_]:= T[n, k]= t[n+1, k+1, 3]; (* t(n, k, 3) = A142458 *)
Flatten[Table[CoefficientList[Sum[T[n, k]*x^k, {k, 0, n}]/(1+x), x], {n, 1, 14, 2}]]
(* 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]]; (* t(n, k, 3) = A142458 *)
A225398[n_, k_]:= A225398[n, k]= Sum[(-1)^(k-j-1)*t[2*n, j+1, 3], {j, 0, k-1}];
|
|
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)
def A225398(n, k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|