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A171229
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A designed integer triangle( a Flajolet-Sedgewick triangle); t(n,k)=If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]]
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4
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 9, 25, 9, 1, 1, 13, 94, 94, 13, 1, 1, 14, 265, 721, 265, 14, 1, 1, 10, 532, 3926, 3926, 532, 10, 1, 1, 5, 739, 14833, 40321, 14833, 739, 5, 1, 1, 2, 701, 38248, 282612, 282612, 38248, 701, 2, 1, 1, 1, 448, 66464, 1334961, 3628801
(list;
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refs;
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history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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Row sums are:
{1, 2, 5, 12, 45, 216, 1281, 8938, 71477, 643128, 6432551,...}.
In several places in Flajolet and Sedgewick,
they claim that the limit
of the Eulerian numbers A008292 is the Gaussian density:
f(x)=Exp[ -x^2/2]/Sqrt[2*Pi]
This sequence is designed to give that Gaussian limiting behavior
in an integer based sequence.
Testing at n=64 level for Sierpinski-ness modulo 2:
Clear[t, n, k, a]
$MaxExtraPrecision = 500
t[n_, k_] = If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]]
a = Table[Table[t[n, k], {k, 0, n}], {n, 0, 64}];
ListDensityPlot[Table[If[m <= n, Mod[a[[n, m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}], Mesh -> False, Frame -> False]
the result is not Sierpinski.
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REFERENCES
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P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 695
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LINKS
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Table of n, a(n) for n=0..60.
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FORMULA
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t(n,k)=If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]]
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EXAMPLE
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{1},
{1, 1},
{1, 3, 1},
{1, 5, 5, 1},
{1, 9, 25, 9, 1},
{1, 13, 94, 94, 13, 1},
{1, 14, 265, 721, 265, 14, 1},
{1, 10, 532, 3926, 3926, 532, 10, 1},
{1, 5, 739, 14833, 40321, 14833, 739, 5, 1},
{1, 2, 701, 38248, 282612, 282612, 38248, 701, 2, 1},
{1, 1, 448, 66464, 1334961, 3628801, 1334961, 66464, 448, 1, 1}
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MATHEMATICA
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Clear[t, n, k, a]
t[n_, k_] = If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]]
a = Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}]
Flatten[a]
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CROSSREFS
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Cf. A008292
Sequence in context: A086620 A137897 A056152 * A125690 A176481 A108553
Adjacent sequences: A171226 A171227 A171228 * A171230 A171231 A171232
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula, Dec 05 2009
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STATUS
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approved
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