A157629 A general recursion triangle with third part a power triangle:m=2; 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, 10, 1, 1, 43, 43, 1, 1, 148, 590, 148, 1, 1, 469, 5018, 5018, 469, 1, 1, 1438, 34047, 91492, 34047, 1438, 1, 1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1, 1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1, 1, 39337
Offset: 0
Examples
{1},
{1, 1},
{1, 10, 1},
{1, 43, 43, 1},
{1, 148, 590, 148, 1},
{1, 469, 5018, 5018, 469, 1},
{1, 1438, 34047, 91492, 34047, 1438, 1},
{1, 4351, 204813, 1187731, 1187731, 204813, 4351, 1},
{1, 13096, 1149652, 12609880, 27234646, 12609880, 1149652, 13096, 1},
{1, 39337, 6188356, 117961172, 478838974, 478838974, 117961172, 6188356, 39337, 1},
{1, 118066, 32448653, 1015124312, 7053594482, 13257922028, 7053594482, 1015124312, 32448653, 118066, 1}
Programs
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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}];
Formula
m=0:Pascal:m=1Eulerian numbers;
m=2;
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).
Comments