|
%I #42 Jan 30 2020 21:29:15
%S 1,1,3,19,193,2721,49171,1084483,28245729,848456353,28875761731,
%T 1098127402131,46150226651233,2124008553358849,106246577894593683,
%U 5739439214861417731,332993721039856822081,20651350143685984386753
%N E.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.
%C Essentially the same as A001517: a(n+1) = A001517(n).
%H W. Mlotkowski, A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=33.2&nrA=19&ppB=401&ppE=408">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%F E.g.f.: exp((1-sqrt(1-4*x))/2).
%F D-finite with recurrence: a(n+2) = 2( 2 n + 1 ) a(n+1) + a(n)
%F Recurrence: y(n+1) = sum(k=0..n, binomial(n, k)*binomial(2k, k)*k!*y(n-k) ).
%F a(1 - n) = a(n). a(n + 1) = A001517(n). - _Michael Somos_, Apr 07 2012
%F G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F a(n) ~ 2^(2*n-3/2)*n^(n-1)/exp(n-1/2). - _Vaclav Kotesovec_, Jun 26 2013
%F a(n) = hypergeom([-n+1, n], [], -1). - _Peter Luschny_, Oct 17 2014
%t y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n],{n,0,17}]
%t With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2],{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 30 2011 *)
%o (PARI) {a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* _Michael Somos_, Apr 07 2012 */
%o (Sage)
%o A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
%o [simplify(A080893(n)) for n in (0..19)] # _Peter Luschny_, Oct 17 2014
%Y Cf. A000108, A001517.
%K easy,nice,nonn
%O 0,3
%A _Emanuele Munarini_, Mar 31 2003
|
