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A144443
Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
8
1, 1, 1, 1, 20, 1, 1, 159, 159, 1, 1, 1138, 4254, 1138, 1, 1, 7997, 77878, 77878, 7997, 1, 1, 56016, 1219167, 2984888, 1219167, 56016, 1, 1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 1, 1, 2745134, 244083268, 2219485106, 4400875078, 2219485106, 244083268, 2745134, 1
OFFSET
1,5
FORMULA
T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*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*(3*n-5)*s(n-1) + 6*s(n-2), s(1) = 1, s(2) = 2.
From G. C. Greubel, Mar 04 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = (1/3)*(10*7^(n-2) - (3*n+1)).
T(n, 3) = (1/18)*(9*n^2 -3*n -11 - 20*(21*n-11)*7^(n-3) + 997*13^(n-3)). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 20, 1;
1, 159, 159, 1;
1, 1138, 4254, 1138, 1;
1, 7997, 77878, 77878, 7997, 1;
1, 56016, 1219167, 2984888, 1219167, 56016, 1;
1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 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, 6, 6], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 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 A144443(n, k): return T(n, k, 6, 6)
flatten([[A144443(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved