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%I #11 Jul 27 2022 06:33:39
%S 1,7,97,2221,75721,3591211,225827617,18168156217,1819029079441,
%T 221716249326991,32313176619313921,5547478498197397477,
%U 1107802527495396486937,254557467773494382397811
%N a(n) = 3F0( -n,-n+1,-n+2;;-1)= n!*(n-1)!* 1F2(-n+2;2,3;-1)/2, where nFm(;;z) are generalized hypergeometric series.
%C Special values of hypergeometric functions.
%F The sequence a(n) can be obtained from the following three generating functions of hypergeometric type:
%F g1(t) = sum(a(n)*t^n/(n!*(n-1)!),n=2..infinity) = (t^2/(1-t))* 1F2(1;2,3;t/(1-t))/2
%F g2(t) = sum(a(n)*t^n/(n!*(n-1)!*(n-2)!), n=2..infinity) = exp(t)*t^2* 0F2(;2,3;t)/2
%F g3(t) = sum(a(n)*t^n/(n!*(n-1)!*(n-2)), n=3..infinity)
%F = t^2*(t/(6*(1-t))* 2F3(1,1;2,3,4;t/(1-t))-log(1-t))/2
%F Note the appearance of the factor (n-2) and not (n-2)! in the denominator of g3.
%F D-finite with recurrence a(n) +(-3*n^2+9*n-7)*a(n-1) +3*(n-1)*(n-3)*(n-2)^2*a(n-2) -(n-1)*(n-4)*(n-2)^2*(n-3)^2*a(n-3)=0. - _R. J. Mathar_, Jul 27 2022
%p A174315 := proc(n)
%p n!*(n-1)!*hypergeom([2-n],[2,3],-1)/2 ;
%p simplify(%) ;
%p end proc:
%p seq(A174315(n),n=2..40) ; # _R. J. Mathar_, Jul 27 2022
%t Table[HypergeometricPFQ[{-n, -n + 1, -n + 2}, {}, -1], {n, 2, 20}] (* _Vaclav Kotesovec_, Jun 08 2021 *)
%K nonn
%O 2,2
%A _Karol A. Penson_ and Katarzyna Gorska (gorska(AT)lptmc.jussieu.fr), Mar 15 2010
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