A174324 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.

1, 4, 31, 391, 7261, 185956, 6271189, 269066701, 14300511481, 921666527596, 70789188893611, 6386088654729499, 668423261212035421, 80325071500899911596, 10981857825124725031081, 1694577083441728891610041

Offset

2,2

Links

Table of n, a(n) for n=2..17.

Formula

The sequence a(n) can be obtained from the following three generating functions of hypergeometric type:

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.

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.

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.

Note the appearance of the factor (n-2) and not (n-2)! in the denominator of g3.

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

Maple

A174324 := proc(n)
    n!*(n-1)!*2^(1-n)*hypergeom([2-n], [2, 3], -2) ;
    simplify(%) ;
end proc:
seq(A174324(n), n=2..40) ; # R. J. Mathar, Jul 27 2022

Mathematica

Table[HypergeometricPFQ[{-n, -n + 1, -n + 2}, {}, -1/2], {n, 2, 20}] (* Vaclav Kotesovec, Jun 08 2021 *)

Crossrefs

Sequence in context: A351798 A005046 A323568 * A211194 A237581 A319074

Adjacent sequences: A174321 A174322 A174323 * A174325 A174326 A174327

Keyword

nonn

Author

Karol A. Penson and Katarzyna Gorska, Mar 15 2010

Status

approved