A157630 A general recursion triangle with third part a power triangle:m=3; Power triangle: f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)]; 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, m)*A(n - 2, k - 1, m).

1, 1, 1, 1, 26, 1, 1, 165, 165, 1, 1, 778, 6054, 778, 1, 1, 3305, 94708, 94708, 3305, 1, 1, 13506, 1042017, 4836404, 1042017, 13506, 1, 1, 54421, 9592365, 133509509, 133509509, 9592365, 54421, 1, 1, 218210, 79849738, 2613951290, 9042784642

Offset

0,5

Comments

Row sums are:

{1, 2, 28, 332, 7612, 196028, 6947452, 286312592, 14430823120, 835866974744,

56181137740936,...}.

Links

Table of n, a(n) for n=0..40.

Formula

m=0;Pascal:m=1;Eulerian numbers;

m=3;

Power triangle:

f(n,k,m)=If[n*k*(n - k) == 0, 1, n^m - (k^m + (n - k)^m)];

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

Example

{1},
{1, 1},
{1, 26, 1},
{1, 165, 165, 1},
{1, 778, 6054, 778, 1},
{1, 3305, 94708, 94708, 3305, 1},
{1, 13506, 1042017, 4836404, 1042017, 13506, 1},
{1, 54421, 9592365, 133509509, 133509509, 9592365, 54421, 1},
{1, 218210, 79849738, 2613951290, 9042784642, 2613951290, 79849738, 218210, 1},
{1, 873513, 625462200, 41642326092, 375664825566, 375664825566, 41642326092, 625462200, 873513, 1},
{1, 3494890, 4714295625, 581099434140, 11320981714506, 32367539862612, 11320981714506, 581099434140, 4714295625, 3494890, 1}

Mathematica

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, m]*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: A177970 A225483 A183065 * A173583 A040687 A040688

Adjacent sequences: A157627 A157628 A157629 * A157631 A157632 A157633

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 03 2009

Status

approved