OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 3) = A002450(n). - G. C. Greubel, Jan 10 2022
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 5, 1;
1, 21, 21, 1;
1, 85, 234, 85, 1;
1, 341, 2110, 2110, 341, 1;
1, 1365, 17163, 35882, 17163, 1365, 1;
1, 5461, 131751, 505979, 505979, 131751, 5461, 1;
1, 21845, 976876, 6395471, 11433118, 6395471, 976876, 21845, 1;
1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1;
MATHEMATICA
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m] + (m*k+1)*T[n-1, k, m] - m*k*(n-k)*T[n-2, k-1, m]];
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)
PROG
(Sage)
@CachedFunction
def T(n, k, m): # A157154
if (k==0 or k==n): return 1
else: return (m*(n-k) +1)*T(n-1, k-1, m) + (m*k+1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 24 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 10 2022
STATUS
approved