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Inverse hyperbinomial transform of A089467.
7

%I #31 Jun 19 2018 12:39:45

%S 1,1,3,18,163,1950,28821,505876,10270569,236644092,6098971555,

%T 173823708696,5427760272507,184267682837992,6757353631762293,

%U 266191329601854000,11210291102456374801,502602430218071545104,23900770928782913595651,1201581698963550283673632

%N Inverse hyperbinomial transform of A089467.

%C See A088956 for the definition of the hyperbinomial transform.

%C a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the functional digraph contains no cycles of length 2. - _Geoffrey Critzer_, Mar 21 2012

%H Vincenzo Librandi, <a href="/A089466/b089466.txt">Table of n, a(n) for n = 0..100</a>

%F A089467(n) = sum(k=0, n, (n-k+1)^(n-k-1)*C(n, k)*a(k)). a(n) = sum(m=0, n, sum(j=0, m, C(m, j)*C(n, n-m-j)*(n-1)^(n-m-j)*(m+j)!/(-2)^j)/m!)).

%F E.g.f.: exp(-(T(x))^2/2)/(1-T(x)), where T(x) is the e.g.f. for A000169. - _Geoffrey Critzer_, Mar 21 2012

%F a(n) ~ exp(-1/2) * n^n. - _Vaclav Kotesovec_, Oct 08 2013

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!). - _Daniel Suteu_, Jun 19 2018

%t nn=20; t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; a=Log[1/(1-t)]; Range[0,nn]! CoefficientList[Series[Exp[a-t^2/2], {x,0,nn}], x] (* _Geoffrey Critzer_, Mar 21 2012 *)

%o (PARI) a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n,n-m-j)*(n-1)^(n-m-j)*(m+j)!/(-2)^j)/m!))

%o (PARI) a(n) = n! * sum(k=0, n\2, (-1/2)^k * n^(n - 2*k) / (k! * (n - 2*k)!)); \\ _Daniel Suteu_, Jun 19 2018

%Y Cf. A089467, A088956.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 08 2003