A174315 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.

1, 7, 97, 2221, 75721, 3591211, 225827617, 18168156217, 1819029079441, 221716249326991, 32313176619313921, 5547478498197397477, 1107802527495396486937, 254557467773494382397811

Offset

2,2

Comments

Special values of hypergeometric functions.

Links

Table of n, a(n) for n=2..15.

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))* 1F2(1;2,3;t/(1-t))/2

g2(t) = sum(a(n)*t^n/(n!*(n-1)!*(n-2)!), n=2..infinity) = exp(t)*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))* 2F3(1,1;2,3,4;t/(1-t))-log(1-t))/2

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

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

Maple

A174315 := proc(n)
    n!*(n-1)!*hypergeom([2-n], [2, 3], -1)/2 ;
    simplify(%) ;
end proc:
seq(A174315(n), n=2..40) ; # R. J. Mathar, Jul 27 2022

Mathematica

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

Crossrefs

Sequence in context: A013521 A003710 A027837 * A046908 A005014 A201063

Adjacent sequences: A174312 A174313 A174314 * A174316 A174317 A174318

Keyword

nonn

Author

Karol A. Penson and Katarzyna Gorska (gorska(AT)lptmc.jussieu.fr), Mar 15 2010

Status

approved