cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A166960 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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Roger L. Bagula, Oct 25 2009

Keywords

Comments

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

Examples

			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}
...

Links

Crossrefs

Cf. A008277, A166961, A166962.

Programs

  • Mathematica
    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 *)

Formula

T(n, k) = (n-k+1)*T(n-1, k-1) + k^2*T(n-1, k).

A166961 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 = 2.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 59, 42, 1, 1, 361, 925, 154, 1, 1, 2175, 16402, 8937, 507, 1, 1, 13061, 265605, 365050, 67500, 1587, 1, 1, 78379, 4127746, 12611845, 5592850, 442242, 4852, 1, 1, 470289, 62935117, 398536866, 365184855, 68337922, 2652742, 14676, 1
Offset: 1

Views

Author

Roger L. Bagula and Mats Granvik, Oct 25 2009

Keywords

Comments

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 A166960, m = 2 yields this sequence, and m = 3 yields A166962. These sequences are, in essence, generalized Stirling numbers of the second kind. - G. C. Greubel, May 29 2016

Examples

			Triangle starts:
{1},
{1, 1},
{1, 9, 1},
{1, 59, 42, 1},
{1, 361, 925, 154, 1},
{1, 2175, 16402, 8937, 507, 1},
{1, 13061, 265605, 365050, 67500, 1587, 1},
{1, 78379, 4127746, 12611845, 5592850, 442242, 4852, 1},
{1, 470289, 62935117, 398536866, 365184855, 68337922, 2652742, 14676, 1},
{1, 2821751, 951081090, 11977188769, 20817224001, 7796966547, 719764976, 15024830, 44181, 1}
...

Links

Crossrefs

Cf. A008277, A166960, A166961.

Programs

  • Mathematica
    A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (2*n - 2*k + 1)*A[n - 1, k - 1] + k*(2*k - 1)*A[n - 1, k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

Formula

T(n,k)= (2*n - 2*k + 1)*T(n - 1, k - 1) + k*(2*k - 1)*T(n - 1, k).
Showing 1-2 of 2 results.

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