cp's OEIS Frontend

A000994 Shifts 2 places left under binomial transform.

%I A000994 M1446 N0572 #61 Jul 03 2023 18:11:04
%S A000994 1,0,1,1,2,5,13,36,109,359,1266,4731,18657,77464,337681,1540381,
%T A000994 7330418,36301105,186688845,995293580,5491595645,31310124067,
%U A000994 184199228226,1116717966103,6968515690273,44710457783760,294655920067105,1992750830574681,13817968813639426
%N A000994 Shifts 2 places left under binomial transform.
%C A000994 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
%D A000994 Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
%D A000994 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000994 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000994 Alois P. Heinz, <a href="/A000994/b000994.txt">Table of n, a(n) for n = 0..650</a> (first 101 terms from T. D. Noe)
%H A000994 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
%H A000994 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A000994 R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H A000994 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A000994 S. Tauber, <a href="http://www.jstor.org/stable/2308653">On generalizations of the exponential function</a>, Amer. Math. Monthly, 67 (1960), 763-767.
%F A000994 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.
%F A000994 However, a(n)/A000995(n) (e.g., 77464/63117) -> 1.228..., the constant in A051148 and A051149.
%F A000994 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
%F A000994 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
%F A000994 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - _Ilya Gutkovskiy_, Aug 09 2020
%e A000994 A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...
%p A000994 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;
%t A000994 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 *)
%o A000994 (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
%o A000994 (Haskell)
%o A000994 a000994 n = a000994_list !! n
%o A000994 a000994_list = 1 : 0 : us where
%o A000994   us = 1 : 1 : f 2 where
%o A000994     f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
%o A000994 -- _Reinhard Zumkeller_, Jun 02 2013
%Y A000994 Cf. A000995, A051139, A051140. Cf. A007318.
%Y A000994 Column k=2 of A143983. Cf. A007476, A088022, A086880.
%K A000994 nonn,easy,nice,eigen
%O A000994 0,5
%A A000994 _N. J. A. Sloane_