OFFSET
0,5
COMMENTS
Row sums are:
{1, 2, 68, 1192, 77216, 3906112, 432940608, 43456890112, 7458742249216,
1239194473427968, 306328813704524800,...}.
FORMULA
m=0;Pascal:m=1;Eulerian numbers;
m=4;
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, 66, 1},
{1, 595, 595, 1},
{1, 3684, 69846, 3684, 1},
{1, 19909, 1933146, 1933146, 19909, 1},
{1, 102246, 32826431, 367083252, 32826431, 102246, 1},
{1, 515671, 437744405, 21290184979, 21290184979, 437744405, 515671, 1},
{1, 2585160, 5091202876, 731091441592, 5986371789958, 731091441592, 5091202876, 2585160, 1},
{1, 12935689, 54443558020, 18708536794676, 600834243425598, 600834243425598, 18708536794676, 54443558020, 12935689, 1},
{1, 64692234, 551304588237, 397892494400440, 35386937552378930, 234758050872405116, 35386937552378930, 397892494400440, 551304588237, 64692234, 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
KEYWORD
AUTHOR
Roger L. Bagula, Mar 03 2009
STATUS
approved