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%I #16 Jul 27 2022 06:37:11
%S 1,4,31,391,7261,185956,6271189,269066701,14300511481,921666527596,
%T 70789188893611,6386088654729499,668423261212035421,
%U 80325071500899911596,10981857825124725031081,1694577083441728891610041
%N a(n) = 3F0(-n,-n+1,-n+2;;-1/2) = n!*(n-1)!*2^(1-n)* 1F2(-n+2;2,3;-2), where nFm(;;) are generalized hypergeometric series.
%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/2))* 1F2(1;2,3;t/(1-t/2))/2.
%F g2(t) = sum(a(n)*t^n/(n!*(n-1)!*(n-2)!), n=2..infinity) = exp(t/2)*t^2* 0F2(;2,3;t)/2.
%F g3(t) = sum(a(n)*t^n/(n!*(n-1)!*(n-2)), n=3..infinity) = t^2*(t/(6*(1-t/2))* 2F3(1,1;2,3,4;t/(1-t/2))-log(1-t/2))/2.
%F Note the appearance of the factor (n-2) and not (n-2)! in the denominator of g3.
%F D-finite with recurrence 8*a(n) +4*(-3*n^2+9*n-8)*a(n-1) +6*(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 A174324 := proc(n)
%p n!*(n-1)!*2^(1-n)*hypergeom([2-n],[2,3],-2) ;
%p simplify(%) ;
%p end proc:
%p seq(A174324(n),n=2..40) ;# _R. J. Mathar_, Jul 27 2022
%t Table[HypergeometricPFQ[{-n, -n + 1, -n + 2}, {}, -1/2], {n, 2, 20}] (* _Vaclav Kotesovec_, Jun 08 2021 *)
%K nonn
%O 2,2
%A _Karol A. Penson_ and Katarzyna Gorska, Mar 15 2010
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