A157453 A general recursion sequence:m=8:Half tent function: f(n,m)== Min[1 + Floor[m/2], 1 + Floor[(n - m)/2]]; Recursion: A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) + (m*k + 1)*A(n - 1, k, m) + m*f(n, k)*A(n - 2, k - 1, m)

1, 1, 1, 1, 2, 1, 1, 19, 19, 1, 1, 188, 630, 188, 1, 1, 1717, 15258, 15258, 1717, 1, 1, 15494, 316047, 762900, 316047, 15494, 1, 1, 139495, 6008053, 29502051, 29502051, 6008053, 139495, 1, 1, 1255520, 109096108, 986409824, 1953238566, 986409824

Offset

0,5

Comments

Row sums are:

{1, 2, 4, 40, 1008, 33952, 1425984, 71299200, 4146761472, 274256650752,

20361340339200,...},

What is unique about this recursion sequence is that it starts off giving sequences near the Eulerian numbers and MacMahon numbers at m=1,2, but at m=7,8 has gone down to a sequence that starts off like the binomial.

Links

Table of n, a(n) for n=0..41.

Formula

m=8:Half tent function:

f(n,m)== Min[1 + Floor[m/2], 1 + Floor[(n - m)/2]];

Recursion: A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) +

(m*k + 1)*A(n - 1, k, m) +

m*f(n, k)* A(n - 2, k - 1, m)

Example

{1},
{1, 1},
{1, 2, 1},
{1, 19, 19, 1},
{1, 188, 630, 188, 1},
{1, 1717, 15258, 15258, 1717, 1},
{1, 15494, 316047, 762900, 316047, 15494, 1},
{1, 139495, 6008053, 29502051, 29502051, 6008053, 139495, 1},
{1, 1255520, 109096108, 986409824, 1953238566, 986409824, 109096108, 1255520, 1},
{1, 11299753, 1927314436, 30054019316, 105135691870, 105135691870, 30054019316, 1927314436, 11299753, 1},
{1, 101697866, 33518917677, 862952943480, 4957907335378, 8652378550396, 4957907335378, 862952943480, 33518917677, 101697866, 1}

Mathematica

Clear[A, a0, b0, n, k, m, f];
f[n_, k_] := Min[1 + Floor[m/2], 1 + Floor[(n - m)/2]];
A[n_, 0, m_] := 1; A[n_, n_, m_] := 1;
A[n_, k_, m_] := (m*(n - k) + 1)*A[n - 1, k - 1, m] + (m*k + 1)*A[n - 1, k, m] + m*f[n, k]*A[n - 2, k - 1, m];
Table[A[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[A[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[A[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Crossrefs

Sequence in context: A179071 A124001 A174966 * A174174 A156889 A172177

Adjacent sequences: A157450 A157451 A157452 * A157454 A157455 A157456

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 01 2009

Status

approved