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A157154
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A general three part recursion triangle sequence second type: m=3; 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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1, 1, 1, 1, 5, 1, 1, 21, 21, 1, 1, 85, 234, 85, 1, 1, 341, 2110, 2110, 341, 1, 1, 1365, 17163, 35882, 17163, 1365, 1, 1, 5461, 131751, 505979, 505979, 131751, 5461, 1, 1, 21845, 976876, 6395471, 11433118, 6395471, 976876, 21845, 1, 1, 87381, 7089360
(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, 44, 406, 4904, 72940, 1286384, 26221504, 606353744, 15680643352,...}.
What I have done here is subtract 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=3;
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, 21, 21, 1},
{1, 85, 234, 85, 1},
{1, 341, 2110, 2110, 341, 1},
{1, 1365, 17163, 35882, 17163, 1365, 1},
{1, 5461, 131751, 505979, 505979, 131751, 5461, 1},
{1, 21845, 976876, 6395471, 11433118, 6395471, 976876, 21845, 1},
{1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1},
{1, 349525, 50761485, 848430732, 3839932578, 6201694710, 3839932578, 848430732, 50761485, 349525, 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: A176242 A036969 A080249 * A022168 A157212 A156600
Adjacent sequences: A157151 A157152 A157153 * A157155 A157156 A157157
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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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