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) = T(n, k).
EXAMPLE
1;
1, 1;
1, 5, 1;
1, 15, 15, 1;
1, 37, 110, 37, 1;
1, 83, 568, 568, 83, 1;
1, 177, 2415, 5534, 2415, 177, 1;
1, 367, 9137, 41027, 41027, 9137, 367, 1;
1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1;
1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1;
MAPLE
A157147:= proc(n, k)
option remember;
if k < 0 or k> n then 0;
elif k = 0 or k = n then 1;
else (n-k+1)*procname(n-1, k-1) +(k+1)*procname(n-1, k) +k*(n-k)*procname(n-2, k-1);
end if;
end proc:
seq(seq(A157147(n, k), k=0..n), n=0..10); # R. J. Mathar, Feb 06 2015
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, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)
PROG
(Sage)
def T(n, k, m): # A157147
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, 1) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 09 2022
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 24 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 09 2022
STATUS
approved