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A000995
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Shifts left two terms under the binomial transform.
(Formerly M1228 N0471)
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21
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0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, 15168, 63117, 275252, 1254801, 5968046, 29551768, 152005634, 810518729, 4472244574, 25497104007, 149993156234, 909326652914, 5674422994544, 36408092349897, 239942657880360
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OFFSET
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0,5
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COMMENTS
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The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - Paul Barry, Oct 06 2007
a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with an ascent (or are empty). For example, a(5)=4 counts 1432, 2314, 2431, 3421. - David Callan, Dec 02 2011
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REFERENCES
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Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n+1)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Oct 28 2006
E.g.f: -2 * exp(x) *( BesselI_0(2) * BesselK_0(2*exp(x/2)) - BesselK_0(2) * 0F1([], [1], exp(x)) ); see the Mathematica program. - Pierre-Louis Giscard, Aug 12 2014
G.f. A(x) satisfies: A(x) = x*(1 + x*A(x/(1 - x))/(1 - x)). - Ilya Gutkovskiy, May 02 2019
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EXAMPLE
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A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...
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MAPLE
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A000995 := proc(n) local k; option remember; if n <= 1 then n else n + add(binomial(n, k)*A000995(k - 2), k = 2 .. n); fi; end;
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MATHEMATICA
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a[n_] := a[n] = If[n <= 1, n, n + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]]; Join[{0, 1}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, May 18 2011, after Maple prog. *)
(* Computation using e.g.f.: *)
nn=20; S=(Series[-2 E^(t/2) Sqrt[E^ t] (BesselI[0, 2] BesselK[0, 2 Sqrt[E^t]] - BesselK[0, 2] Hypergeometric0F1[1, E^t]), {t, 0, nn}]); Flatten[{0, 1, FullSimplify[Table[CoefficientList[Normal[S], t][[i]] (i - 1)!, {i, 1, nn}]]}] (* Pierre-Louis Giscard, Aug 12 2014 *)
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PROG
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(PARI) a(n)=polcoeff(sum(k=0, n, x^(2*k+1)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Oct 28 2006
(Haskell)
a000995 n = a000995_list !! n
a000995_list = 0 : 1 : vs where
vs = 0 : 1 : g 2 where
g x = (x + sum (zipWith (*) (map (a007318' x) [2..x]) vs)) : g (x + 1)
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CROSSREFS
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KEYWORD
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nonn,eigen,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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