A000995 Shifts left two terms under the binomial transform.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• eigen
• easy
• nice

Comments

The binomial transform of A000995 has g.f. x*c(x)^2/(1+x^2*c(x)^2). - Paul Barry, Oct 06 2007

Equals row sums of triangle A137854 such that A000995(3) = 1 = first row of triangle A137854. - Gary W. Adamson, Feb 15 2008

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

Examples

			A(x) = x + x^3/(1-x)^2 + x^5/((1-x)*(1-2x))^2 + x^7/((1-x)*(1-2x)*(1-3x))^2 +...
		

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

• T. D. Noe, Table of n, a(n) for n = 0..100
• S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Crossrefs

Cf. A000994, A051139, A051140.

Cf. A137854.

Cf. A007318.

Programs

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)
-- Reinhard Zumkeller, Jun 02 2013

Maple

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;

Mathematica

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 *)

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

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)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Oct 28 2006

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

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

Extensions

More terms from Paul D. Hanna, Oct 28 2006