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.

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

1,5

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

Links

G. C. Greubel, Table of n, a(n) for the first 25 rows

Formula

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

Example

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}
...

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 *)

Crossrefs

Cf. A008277, A166960, A166961.

Sequence in context: A359313 A304321 A156278 * A202988 A098436 A022172

Adjacent sequences: A166958 A166959 A166960 * A166962 A166963 A166964

Keyword

nonn,tabl

Author

Roger L. Bagula and Mats Granvik, Oct 25 2009

Status

approved