A066399 - OEIS

%I #46 Apr 07 2019 19:04:21

%S 0,1,-4,39,-616,13505,-379296,12995983,-525688192,24519144609,

%T -1295527513600,76481653648631,-4989249262503936,356408413864589281,

%U -27670449142629400576,2319870547729387929375,-208886312501433616531456,20104397299878424990749377

%N From reversion of e.g.f. for squares.

%H Robert Israel, <a href="/A066399/b066399.txt">Table of n, a(n) for n = 0..300</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n+1) = (-1)^n*(n)! * Sum_{m=0..n} (n+1)^m/m! * binomial(2*n-m,n). - _Vladimir Kruchinin_, Feb 22 2011

%F For n>=2, a(n) = (-2)^(n-1)*(2n-3)!!*hypergeom([1-n], [2-2n], n), where n!! denotes the double factorial A006882. - _Vladimir Reshetnikov_, Oct 16 2015

%F E.g.f. g(x) satisfies (g(x) + g(x)^2)*exp(g(x)) = x. - _Robert Israel_, Oct 16 2015

%F a(n) ~ (-1)^(n-1) * (2 + sqrt(5))^(n-1/2) * n^(n-1) / (5^(1/4) * exp((sqrt(5) - 1)*n/2)). - _Vaclav Kotesovec_, Oct 18 2015

%p read transforms; add(n^2*x^n/n!,n=1..30); series(%,x,31): seriestoseries(%,'revogf'); SERIESTOLISTMULT(%);

%p with(powseries):powcreate(t(n)=n^2/n!):seq(n!*coeff(tpsform(reversion(t),x,19),x,n),n=0..18); spec:=[A,{A=Prod(Z,Set(A),Set(B)),B=Cycle(A)},labeled];seq(combstruct[count](spec,size=n), n=0..18); # _Vladeta Jovovic_, May 29 2007

%p a := n -> `if`(n<2,n,(-2)^(n-1)*doublefactorial(2*n-3)*hypergeom([1-n],[2-2*n],n)): seq(simplify(a(n)),n=0..18); # _Peter Luschny_, Oct 16 2015

%t A066399[0] = 0; A066399[1] = 1; A066399[n_] := (-2)^(n - 1) (2 n - 3)!! Hypergeometric1F1[1 - n, 2 - 2 n, n]; Table[A066399[n], {n, 0, 10}] (* _Vladimir Reshetnikov_, Oct 16 2015 *)

%o (PARI) a(n) = if(n==0, 0, (-1)^(n-1)*(n-1)! * sum(k=0, n-1, (n)^k/k! * binomial(2*n-2-k,n-1))) \\ _Altug Alkan_, Oct 16 2015

%Y Cf. A295188.

%K sign

%O 0,3

%A _N. J. A. Sloane_, Dec 25 2001