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A000994
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Shifts 2 places left under binomial transform.
(Formerly M1446 N0572)
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8
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1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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 a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011
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REFERENCES
| 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).
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
N. J. A. Sloane, Transforms
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FORMULA
| 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.
However, A000994(n)/A000995(n) [ e.g. 77464/63117 ] -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n)*(1-n*x)/Product_{k=0..n} (1-k*x)^2 . - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 02 2006
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EXAMPLE
| A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...
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MAPLE
| A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;
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MATHEMATICA
| a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* From Jean-François Alcover, Oct 11 2011, after Maple *)
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PROG
| (PARI) a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 02 2006
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CROSSREFS
| Cf. A000995, A051139, A051140.
Sequence in context: A133365 A135335 A066723 * A148296 A148297 A148298
Adjacent sequences: A000991 A000992 A000993 * A000995 A000996 A000997
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KEYWORD
| nonn,easy,nice,eigen
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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