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A009578 E.g.f. sinh(log(1+x))/exp(x). Unsigned sequence gives degrees of (finite by nilpotent) representations of Braid groups.

%I A009578 #38 Aug 15 2025 05:41:50
%S A009578 0,1,-3,9,-34,165,-981,6853,-54804,493209,-4932055,54252561,
%T A009578 -651030678,8463398749,-118487582409,1777313736045,-28437019776616,
%U A009578 483429336202353,-8701728051642219,165332832981202009,-3306656659624040010
%N A009578 E.g.f. sinh(log(1+x))/exp(x). Unsigned sequence gives degrees of (finite by nilpotent) representations of Braid groups.
%C A009578 Unsigned sequence gives number of undirected paths through K_n, and including paths with zero edges. - _Thomas Anton_, Jan 14 2021
%H A009578 G. C. Greubel, <a href="/A009578/b009578.txt">Table of n, a(n) for n = 0..448</a>
%F A009578 a(n) = n*(1 + A000522(n-1))/2 for n > 0.
%F A009578 For the unsigned sequence, a(n) = n*(2*a(n-1) + 3 - n)/2, a(0)=0. - _Emeric Deutsch_, Oct 24 2001
%F A009578 a(n) = (-1)^(n-1)/2*floor(n!*exp(1) + n-1), n > 0. - _Vladeta Jovovic_, Aug 17 2002
%F A009578 The defining generating function simplifies to x*(2+x)*exp(-x)/(2*(1+x)). - _Emeric Deutsch_, Nov 16 2007
%F A009578 E.g.f.: 1/2*U(0) where U(k)= 1 + x/(1 + 1/(x*(x+1) - 1 - x^2*(x+1)^2/(x*(x+1) + (k+1)/U(k+1)))); (continued fraction, 3rd kind, 4-step). - _Sergei N. Gladkovskii_, Nov 16 2012
%F A009578 D-finite with recurrence a(n) +(n+3)*a(n-1) +3*n*a(n-2) +(3*n-5)*a(n-3) +(n-3)*a(n-4)=0. - _R. J. Mathar_, Aug 15 2025
%p A009578 g:=(1/2)*x*(2+x)*exp(-x)/(1+x): gser:=series(g,x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # _Emeric Deutsch_, Nov 16 2007
%t A009578 With[{nmax=30}, CoefficientList[Series[Sinh[Log[1+x]]*Exp[-x], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Jan 21 2018 *)
%o A009578 (Magma) f := function(n) nn := n; for i := 2 to n do nn := nn+Factorial(n)/(Factorial(n-i)*2); end for; return nn; end function;
%o A009578 (PARI) my(x='x+O('x^30)); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(-x) ))) \\ _G. C. Greubel_, Jan 21 2018
%Y A009578 Cf. A009132.
%K A009578 sign,easy
%O A009578 0,3
%A A009578 _R. H. Hardin_, _Stephen P. Humphries_
%E A009578 Extended with signs by _Olivier Gérard_, Mar 15 1997