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A157147
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A general three part recursion triangle sequence: 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*k*(n - k)*A(n - 2, k - 1, m).
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0
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1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 110, 37, 1, 1, 83, 568, 568, 83, 1, 1, 177, 2415, 5534, 2415, 177, 1, 1, 367, 9137, 41027, 41027, 9137, 367, 1, 1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1, 1, 1515, 107442, 1409814, 4845540, 4845540, 1409814
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| The row sums are:
{1, 2, 7, 32, 186, 1304, 10720, 101064, 1074832, 12728624, 166105008,...}.
What I have done here is add a new symmetrical part
to the "zero start" Sierpinski -Pascal recursion at "down two" or n-2 in my notation:
m*k*(n - k)*A(n - 2, k - 1, m).
It uses the symmetrical k*(n-k) multiplier.
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FORMULA
| 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*k*(n - k)*A(n - 2, k - 1, m).
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EXAMPLE
| {1},
{1, 1},
{1, 5, 1},
{1, 15, 15, 1},
{1, 37, 110, 37, 1},
{1, 83, 568, 568, 83, 1},
{1, 177, 2415, 5534, 2415, 177, 1},
{1, 367, 9137, 41027, 41027, 9137, 367, 1},
{1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1},
{1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1},
{1, 3049, 347945, 7172976, 40220118, 70616830, 40220118, 7172976, 347945, 3049, 1}
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MATHEMATICA
| Clear[A, n, k, m];
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*k*(n - k)*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}]
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CROSSREFS
| Sequence in context: A168288 A157523 A141691 * A156920 A174044 A174159
Adjacent sequences: A157144 A157145 A157146 * A157148 A157149 A157150
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 24 2009
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