OFFSET
1,5
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*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) = 2*(2*n-3)*s(n-1) + 4*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) = (1/2)*(7*5^(n-2) - (2*n+1)).
T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 14, 1;
1, 83, 83, 1;
1, 432, 1550, 432, 1;
1, 2181, 19898, 19898, 2181, 1;
1, 10930, 217887, 523548, 217887, 10930, 1;
1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1;
1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 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, 4, 4], {n, 15}, {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 A144441(n, k): return T(n, k, 4, 4)
flatten([[A144441(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