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A144445
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Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
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8
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1, 1, 1, 1, 23, 1, 1, 206, 206, 1, 1, 1677, 6341, 1677, 1, 1, 13452, 133451, 133451, 13452, 1, 1, 107659, 2403612, 5916231, 2403612, 107659, 1, 1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1, 1, 6890633, 638151479, 5875203446, 11687580863, 5875203446, 638151479, 6890633, 1
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OFFSET
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1,5
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LINKS
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FORMULA
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T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*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) = (7*n-12)*s(n-1) + 7*s(n-2), s(1) = 1, s(2) = 2.
T(n, n-k) = T(n, k).
T(n, 2) = (1/7)*(23*8^(n-2) - (7*n+2)).
T(n, 3) = (1/98)*(49*n^2 - 21*n - 59 - 46*(56*n-33)*8^(n-3) + 5989*15^(n-3)). (End)
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 23, 1;
1, 206, 206, 1;
1, 1677, 6341, 1677, 1;
1, 13452, 133451, 133451, 13452, 1;
1, 107659, 2403612, 5916231, 2403612, 107659, 1;
1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1;
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MATHEMATICA
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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, 7, 7], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 2022 *)
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PROG
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(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 A144445(n, k): return T(n, k, 7, 7)
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CROSSREFS
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Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144444.
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KEYWORD
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AUTHOR
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STATUS
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approved
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