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A166960
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Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.
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3
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1, 1, 1, 1, 6, 1, 1, 27, 21, 1, 1, 112, 270, 58, 1, 1, 453, 2878, 1738, 141, 1, 1, 1818, 28167, 39320, 8739, 318, 1, 1, 7279, 264411, 769955, 375755, 37665, 685, 1, 1, 29124, 2430652, 13905746, 13243650, 2858960, 146560, 1434, 1, 1, 116505, 22108860
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OFFSET
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1,5
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COMMENTS
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The general recursion relation T(n,k)= (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) connects several sequences for differing values of m. These are: m = 0 yields A008277, m = 1 yields this sequence, m = 2 yields A166961, and m = 3 yields A166962. These sequences are, in essence, generalized Stirling numbers of the second kind. - G. C. Greubel, May 29 2016
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LINKS
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FORMULA
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T(n, k) = (n-k+1)*T(n-1, k-1) + k^2*T(n-1, k).
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EXAMPLE
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Triangle starts:
{1},
{1, 1},
{1, 6, 1},
{1, 27, 21, 1},
{1, 112, 270, 58, 1},
{1, 453, 2878, 1738, 141, 1},
{1, 1818, 28167, 39320, 8739, 318, 1},
{1, 7279, 264411, 769955, 375755, 37665, 685, 1},
{1, 29124, 2430652, 13905746, 13243650, 2858960, 146560, 1434, 1},
{1, 116505, 22108860, 239506500, 414525726, 169140810, 18617280, 531456, 2949, 1}
...
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MATHEMATICA
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A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (n - k + 1)*A[n - 1, k - 1] + k^2*A[n - 1, k]; Flatten[Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)
T[ n_, k_] := Which[k < 1 || k > n, 0, 1 == k == n, 1, True, T[n, k] = k^2 T[n - 1, k] + (n - k + 1) T[n - 1, k - 1]]; (* Michael Somos, Apr 12 2019 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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