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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
1, 7, 97, 2221, 75721, 3591211, 225827617, 18168156217, 1819029079441, 221716249326991, 32313176619313921, 5547478498197397477, 1107802527495396486937, 254557467773494382397811 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 01:35 EST 2022. Contains 358421 sequences. (Running on oeis4.)