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*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 3.
T(n, n-k, m) = T(n, k, m).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 11, 1;
1, 54, 54, 1;
1, 229, 822, 229, 1;
1, 932, 8368, 8368, 932, 1;
1, 3747, 72066, 174758, 72066, 3747, 1;
1, 15010, 570006, 2759750, 2759750, 570006, 15010, 1;
1, 60065, 4297714, 37366190, 73850596, 37366190, 4297714, 60065, 1;
MATHEMATICA
f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
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*f[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)
def f(n, k): return k if (k <= n//2) else n-k
@CachedFunction
def T(n, k, m): # A157209
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*f(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
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 25 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 10 2022
STATUS
approved