A000994 Shifts 2 places left under binomial transform.

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, 1992750830574681, 13817968813639426

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• eigen

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

Examples

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...
		

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).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
• N. J. A. Sloane, Transforms
• S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Crossrefs

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

Programs

Haskell

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

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;

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}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

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, Nov 02 2006

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, a(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, Nov 02 2006

Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015

G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020