A066399 From reversion of e.g.f. for squares.

0, 1, -4, 39, -616, 13505, -379296, 12995983, -525688192, 24519144609, -1295527513600, 76481653648631, -4989249262503936, 356408413864589281, -27670449142629400576, 2319870547729387929375, -208886312501433616531456, 20104397299878424990749377

Offset: 0

Author

N. J. A. Sloane, Dec 25 2001

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..300
• Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
• Index entries for reversions of series

Crossrefs

Cf. A295188.

Programs

Maple

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

Mathematica

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 *)

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

Formula

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

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

E.g.f. g(x) satisfies (g(x) + g(x)^2)*exp(g(x)) = x. - Robert Israel, Oct 16 2015

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