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A156594
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A q-Stirling 2nd triangle sequence:q=3;m=2; t(n,k)=If[m == 0, n!, Product[Sum[(-1)^i*StirlingS2[ k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
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1, 1, 1, 1, -3, 1, 1, 6, 6, 1, 1, -3, 6, -3, 1, 1, -21, -21, -21, -21, 1, 1, 24, 168, -84, 168, 24, 1, 1, 195, -1560, 5460, 5460, -1560, 195, 1, 1, -111, 7215, 28860, 202020, 28860, 7215, -111, 1, 1, -3072, -113664, -3694080, 29552640, 29552640, -3694080
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:
{1, 2, -1, 14, 2, -82, 302, 8192, 273950, 51483650, -4132493224,...}.
On the sequence only q=2 and q=3 are Integers,
the rest have a few rational terms.
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FORMULA
| q=3;m=2;
t(n,k)=If[m == 0, n!, Product[Sum[(-1)^i*StirlingS2[ k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].
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EXAMPLE
| {1},
{1, 1},
{1, -3, 1},
{1, 6, 6, 1},
{1, -3, 6, -3, 1},
{1, -21, -21, -21, -21, 1},
{1, 24, 168, -84, 168, 24, 1},
{1, 195, -1560, 5460, 5460, -1560, 195, 1},
{1, -111, 7215, 28860, 202020, 28860, 7215, -111, 1},
{1, -3072, -113664, -3694080, 29552640, 29552640, -3694080, -113664, -3072, 1},
{1, -4053, -4150272, 76780032, -4990702080, 5703659520, -4990702080, 76780032, -4150272, -4053, 1}
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MATHEMATICA
| t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^i* StirlingS2[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = f[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
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CROSSREFS
| Sequence in context: A159572 A190907 A035582 * A109647 A176668 A054120
Adjacent sequences: A156591 A156592 A156593 * A156595 A156596 A156597
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 10 2009
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