OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
Sum_{k=1..n} T(n, k) = s(n), where s(n) = (5*n-8)*s(n-1) + 5*s(n-2), with s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/5)*(17*6^(n - 2) - (5*n + 2)).
T(n, 3) = (1/50)*(25*n^2 - 5*n - 31 - 34*6^(n - 3)*(30*n - 13) +
2489*11^(n - 3)). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 17, 1;
1, 118, 118, 1;
1, 729, 2681, 729, 1;
1, 4400, 41745, 41745, 4400, 1;
1, 26431, 555240, 1349245, 555240, 26431, 1;
1, 158622, 6816846, 33456685, 33456685, 6816846, 158622, 1;
1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 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, 5, 5], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 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 A144442(n, k): return T(n, k, 5, 5)
flatten([[A144442(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 05 2008
STATUS
approved