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 A171229 A designed integer triangle( a Flajolet-Sedgewick triangle); t(n,k)=If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]] 4
 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; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. REFERENCES P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 695 LINKS FORMULA t(n,k)=If[n == 0, 1, 1 + Floor[n!*Exp[ -(k - Floor[n]/2)^2]]] EXAMPLE {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} MATHEMATICA 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] CROSSREFS Cf. A008292 Sequence in context: A086620 A137897 A056152 * A125690 A176481 A108553 Adjacent sequences:  A171226 A171227 A171228 * A171230 A171231 A171232 KEYWORD nonn,uned,tabl AUTHOR Roger L. Bagula, Dec 05 2009 STATUS approved

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