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) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 1.
T(n, n-k, m) = T(n, k, m).
T(n, 1, 1) = A101945(n-1), for n >= 1. - G. C. Greubel, Feb 05 2022
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 17, 17, 1;
1, 40, 126, 40, 1;
1, 87, 606, 606, 87, 1;
1, 182, 2413, 5604, 2413, 182, 1;
1, 373, 8679, 38117, 38117, 8679, 373, 1;
1, 756, 29376, 219020, 426002, 219020, 29376, 756, 1;
1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1;
MATHEMATICA
f[n_, k_]:= If[k<=Floor[n/2], 2*k, 2*(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, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 05 2022 *)
PROG
(Sage)
def f(n, k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n, k, m): # A157275
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..12)]) # G. C. Greubel, Feb 05 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 26 2009
EXTENSIONS
Edited by G. C. Greubel, Feb 05 2022
STATUS
approved