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A000995 Shifts left two terms under the binomial transform.

%I A000995 M1228 N0471 #70 May 02 2019 08:50:18
%S A000995 0,1,0,1,2,4,10,29,90,295,1030,3838,15168,63117,275252,1254801,
%T A000995 5968046,29551768,152005634,810518729,4472244574,25497104007,
%U A000995 149993156234,909326652914,5674422994544,36408092349897,239942657880360
%N A000995 Shifts left two terms under the binomial transform.
%C A000995 The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - _Paul Barry_, Oct 06 2007
%C A000995 Equals row sums of triangle A137854 such that A000995(3) = 1 = first row of triangle A137854. - _Gary W. Adamson_, Feb 15 2008
%C A000995 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
%D A000995 Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
%D A000995 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000995 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000995 T. D. Noe, <a href="/A000995/b000995.txt">Table of n, a(n) for n = 0..100</a>
%H A000995 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 A000995 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 A000995 However, A000994(n)/A000995(n) [ e.g., 77464/63117 ] -> 1.228..., the constant in A051148 and A051149.
%F A000995 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
%F A000995 G.f.: (1+2*x^2*c(x)^2)/(1+x^2*c(x^2)), c(x) the g.f. of A000108. - _Paul Barry_, Oct 06 2007. This g.f. is incorrect. - _Vaclav Kotesovec_, Aug 14 2014
%F A000995 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
%F A000995 G.f. A(x) satisfies: A(x) = x*(1 + x*A(x/(1 - x))/(1 - x)). - _Ilya Gutkovskiy_, May 02 2019
%e A000995 A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...
%p A000995 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;
%t A000995 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. *)
%t A000995 (* Computation using e.g.f.: *)
%t A000995 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 *)
%o A000995 (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
%o A000995 (Haskell)
%o A000995 a000995 n = a000995_list !! n
%o A000995 a000995_list = 0 : 1 : vs where
%o A000995   vs = 0 : 1 : g 2 where
%o A000995     g x = (x + sum (zipWith (*) (map (a007318' x) [2..x]) vs)) : g (x + 1)
%o A000995 -- _Reinhard Zumkeller_, Jun 02 2013
%Y A000995 Cf. A000994, A051139, A051140.
%Y A000995 Cf. A137854.
%Y A000995 Cf. A007318.
%K A000995 nonn,eigen,easy,nice
%O A000995 0,5
%A A000995 _N. J. A. Sloane_
%E A000995 More terms from _Paul D. Hanna_, Oct 28 2006