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A144436
Triangle 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 = 1, and j = 4, read by rows.
11
1, 1, 1, 1, 8, 1, 1, 23, 23, 1, 1, 54, 170, 54, 1, 1, 117, 818, 818, 117, 1, 1, 244, 3255, 7224, 3255, 244, 1, 1, 499, 11697, 48443, 48443, 11697, 499, 1, 1, 1010, 39560, 276974, 513326, 276974, 39560, 1010, 1, 1, 2033, 128756, 1431604, 4422246
OFFSET
1,5
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 = 1, and j = 4.
From G. C. Greubel, Mar 03 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 2) = 2^(n+1) - (n+5).
T(n, 3) = (1/2)*( n^2 + 9*n + 16 - 2^(n+2)*(n+3) + 142*3^(n-3) ). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 23, 23, 1;
1, 54, 170, 54, 1;
1, 117, 818, 818, 117, 1;
1, 244, 3255, 7224, 3255, 244, 1;
1, 499, 11697, 48443, 48443, 11697, 499, 1;
1, 1010, 39560, 276974, 513326, 276974, 39560, 1010, 1;
1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 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, 1, 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 A144436(n, k): return T(n, k, 1, 4)
flatten([[A144436(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2022
STATUS
approved