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 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). 0

%I

%S 1,1,1,1,26,1,1,165,165,1,1,778,6054,778,1,1,3305,94708,94708,3305,1,

%T 1,13506,1042017,4836404,1042017,13506,1,1,54421,9592365,133509509,

%U 133509509,9592365,54421,1,1,218210,79849738,2613951290,9042784642

%N 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).

%C Row sums are:

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

%C 56181137740936,...}.

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

%F m=3;

%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, 26, 1},

%e {1, 165, 165, 1},

%e {1, 778, 6054, 778, 1},

%e {1, 3305, 94708, 94708, 3305, 1},

%e {1, 13506, 1042017, 4836404, 1042017, 13506, 1},

%e {1, 54421, 9592365, 133509509, 133509509, 9592365, 54421, 1},

%e {1, 218210, 79849738, 2613951290, 9042784642, 2613951290, 79849738, 218210, 1},

%e {1, 873513, 625462200, 41642326092, 375664825566, 375664825566, 41642326092, 625462200, 873513, 1},

%e {1, 3494890, 4714295625, 581099434140, 11320981714506, 32367539862612, 11320981714506, 581099434140, 4714295625, 3494890, 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 February 26 03:29 EST 2020. Contains 332273 sequences. (Running on oeis4.)