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A144442
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Triangle read by rows: 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.
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8
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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
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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) = (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.
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)
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EXAMPLE
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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;
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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, 5, 5], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 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 A144442(n, k): return T(n, k, 5, 5)
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CROSSREFS
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Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144443, A144444, A144445.
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KEYWORD
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AUTHOR
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STATUS
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approved
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