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A156540
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An anti-diagonal triangular sequence from the "blended" q-factorial: t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]].
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0
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1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 21, 24, 1, 1, 5, 52, 315, 120, 1, 1, 6, 105, 2080, 9765, 720, 1, 1, 7, 186, 8925, 251680, 615195, 5040, 1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320, 1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row sums are:
{1, 2, 4, 11, 51, 494, 12678, 881035, 172854357, 124231859362,
361386799988240}.
Here q=1 is defined as n! instead of being undefined.
This method allows a smooth transition from traditional combinations
to the higher level ones.
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FORMULA
| t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
a(n,m)_out=Anti-diagonal(t(n,m))
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EXAMPLE
| {1},
{1, 1},
{1, 1, 2},
{1, 1, 3, 6},
{1, 1, 4, 21, 24},
{1, 1, 5, 52, 315, 120},
{1, 1, 6, 105, 2080, 9765, 720},
{1, 1, 7, 186, 8925, 251680, 615195, 5040},
{1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320},
{1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880},
{1, 1, 10, 657, 182400, 121226245, 88515803376, 22686496457625, 328430963660800, 10180699028325, 3628800}
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MATHEMATICA
| Clear[t, n, m, i, k, a];
t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];
b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}];
Flatten[%]
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CROSSREFS
| Sequence in context: A174480 A111670 A123353 * A156582 A156953 A156881
Adjacent sequences: A156537 A156538 A156539 * A156541 A156542 A156543
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 09 2009
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