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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
OFFSET
1,3
FORMULA
From G. C. Greubel, Mar 19 2022: (Start)
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}];
Table[A225398[n, k], {n, 12}, {k, 2*n-1}] //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)
def A225398(n, k): return sum( (-1)^(k-j-1)*A142458(2*n, j+1) for j in (0..k-1) )
flatten([[A225398(n, k) for k in (1..2*n-1)] for n in (1..12)]) # G. C. Greubel, Mar 19 2022
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Apr 26 2013 (Entered by N. J. A. Sloane, May 06 2013)
EXTENSIONS
Edited by N. J. A. Sloane, May 11 2013
STATUS
approved