a PolyGamma sequence and a addition to A002720

[Wouter Meeussen]

in Mathematica 3.0 :

Table[(PolyGamma[m]+EulerGamma) (m-1)!,{m,1,24}]
{0,1,3,11,50,274,1764,13068,109584,1026576,10628640,120543840,1486442880,
  19802759040,283465647360,4339163001600,70734282393600,1223405590579200,
  22376988058521600,431565146817638400,8752948036761600000,
  186244810780170240000,4148476779335454720000,96538966652493066240000}
 
EIS says :
" The terms
" 0,1,3,11,50,274,1764,13068,109584 
" do not match anything in the table...

so, maybe :

%I A000000 
%S A000000 0,1,3,11,50,274,1764,13068,109584,1026576,10628640,120543840,
%T A000000 1486442880,19802759040,283465647360,4339163001600,70734282393600,
%U A000000 1223405590579200,22376988058521600,431565146817638400
%N A000000 the logarithmic derivative of the gamma function at integer
argument, with EulerGamma offset
%R A000000 
%A A000000 w.meeussen.vdmcc at vandemoortele.be
%O A000000 0,3
%K A000000 nonn


**************************************************************************

the EIS *does* contain:

Table[n!  LaguerreL[n,-1],{n,0,12}]
{1,2,7,34,209,1546,13327,130922,1441729,17572114,234662231,3405357682,
  53334454417}

as:

Matches (up to a limit of 10) found for  1 2 7 34 209 1546 13327 130922
1441729  :

- ----------------------
%I A002720 M1795 N0708
%S A002720 1,2,7,34,209,1546,13327,130922,1441729,17572114,234662231,3405357682,
%T A002720
53334454417,896324308634,16083557845279,306827170866106,6199668952527617
%N A002720 Partial permutations of an n-set; number of n X n binary matrices
with at most one 1 in each row and column.
%R A002720 SE33 78.
%O A002720 0,2
%A A002720 njas
%F A002720 a(n) = Sum k!C(n,k)^2, k=0..n. Egf: (1/(1-x))*exp(x/(1-x)).
Recurrence: a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).
%K A002720 nonn,easy
%E A002720 Egf from D. E. Knuth 7/95. Second description from R. H. Hardin
(rhh at research.att.com) 11/97.

- -----------------------

why supplement it ? well, the recursion formula

In[]:=  Sum[ k!Binomial[n,k]^2,{k,0,n}]
Out[]=  (-1)^n*HypergeometricU[-n, 1, -1]

so we ought to be able to simplify 
(-1)^n*HypergeometricU[-n, 1, -1]   to    n!*LaguerreL[n, -1] 
but Mathematica 3.0.0 doesn't do that.

Also, the recursion solver (DiscreteMath..) RSolve comes up empty with :

RSolve[{ a[n] == 2n*a[n-1] - (n-1)^2*a[n-2],a[0]==1,a[1]==2},a[n],n]

this gives me the impression that the "n!*LaguerreL[n, -1]" is not a
self-evident transformation of the known stuff. But maybe I'm out of my
leage here.

the exponential generating function  (1/(1-x))*E^(x/(1-x)) leads to the
conclusion that, over the interval x=[0,1[ we can (surprisingly) set :

  Sum[  LaguerreL[n, -1] x^n , {n,0,Infinity}] == (1/(1-x))*E^(x/(1-x))


I enjoyed it, hope you do too,

wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be