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 = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A094002(n-1). - G. C. Greubel, Jan 10 2022
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 5, 1;
1, 14, 14, 1;
1, 33, 94, 33, 1;
1, 72, 442, 442, 72, 1;
1, 151, 1752, 3818, 1752, 151, 1;
1, 310, 6306, 25358, 25358, 6306, 310, 1;
1, 629, 21390, 144524, 268852, 144524, 21390, 629, 1;
1, 1268, 69822, 746744, 2312836, 2312836, 746744, 69822, 1268, 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, 1], {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): # A157207
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, 1) 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