OFFSET
1,3
LINKS
G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened
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 *)
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)
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