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 A157631 A general recursion triangle with third part a power triangle: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). 0

%I

%S 1,1,1,1,66,1,1,595,595,1,1,3684,69846,3684,1,1,19909,1933146,1933146,

%T 19909,1,1,102246,32826431,367083252,32826431,102246,1,1,515671,

%U 437744405,21290184979,21290184979,437744405,515671,1,1,2585160

%N A general recursion triangle with third part a power triangle: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).

%C Row sums are:

%C {1, 2, 68, 1192, 77216, 3906112, 432940608, 43456890112, 7458742249216,

%C 1239194473427968, 306328813704524800,...}.

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

%F m=4;

%F Power triangle:

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

%F Recursion:

%F A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) +

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

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

%e {1},

%e {1, 1},

%e {1, 66, 1},

%e {1, 595, 595, 1},

%e {1, 3684, 69846, 3684, 1},

%e {1, 19909, 1933146, 1933146, 19909, 1},

%e {1, 102246, 32826431, 367083252, 32826431, 102246, 1},

%e {1, 515671, 437744405, 21290184979, 21290184979, 437744405, 515671, 1},

%e {1, 2585160, 5091202876, 731091441592, 5986371789958, 731091441592, 5091202876, 2585160, 1},

%e {1, 12935689, 54443558020, 18708536794676, 600834243425598, 600834243425598, 18708536794676, 54443558020, 12935689, 1},

%e {1, 64692234, 551304588237, 397892494400440, 35386937552378930, 234758050872405116, 35386937552378930, 397892494400440, 551304588237, 64692234, 1}

%t A[n_, 0, m_] := 1; A[n_, n_, m_] := 1;

%t 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];

%t Table[A[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];

%t Table[Flatten[Table[Table[A[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]

%t Table[Table[Sum[A[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

%K nonn,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Mar 03 2009

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Last modified January 27 12:01 EST 2020. Contains 331295 sequences. (Running on oeis4.)