OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2.
Sum_{k=0..n} T(n, k) = s(n), where s(n) = 2*(n-1)*s(n-1) + 2*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = 4*3^(n-2) - (n+1).
T(n, 3) = (1/2)*(71*5^(n-3) - 8*(3*n+1)*3^(n-3) + n^2 + n - 1). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 31, 31, 1;
1, 102, 326, 102, 1;
1, 317, 2406, 2406, 317, 1;
1, 964, 15087, 34336, 15087, 964, 1;
1, 2907, 86673, 380947, 380947, 86673, 2907, 1;
1, 8738, 473084, 3650206, 6925718, 3650206, 473084, 8738, 1;
1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;
MATHEMATICA
T[n_, k_, m_, j_]:= T[n, k, m, j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m, j] + (m*(k-1)+1)*T[n-1, k, m, j] + j*T[n-2, k-1, m, j]];
Table[T[n, k, 2, 2], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 10 2022 *)
PROG
(Sage)
def T(n, k, m, j):
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*T(n-1, k-1, m, j) + (m*(k-1)+1)*T(n-1, k, m, j) + j*T(n-2, k-1, m, j)
def A144439(n, k): return T(n, k, 2, 2)
flatten([[A144439(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 10 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 05 2008
STATUS
approved