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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 28, 332, 7612, 196028, 6947452, 286312592, 14430823120, 835866974744, 56181137740936,...}. LINKS 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

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Last modified January 26 01:48 EST 2020. Contains 331270 sequences. (Running on oeis4.)