# A062817: Self-Convolution Square-Root and Asymptotics

Paul D. Hanna pauldhanna at juno.com
Sat Oct 6 07:38:29 CEST 2007

```Seqfans,
Consider the sequence:
A062817(n) = Sum_{k=0..n} (n-k)^k*k^(n-k),  n>=1
0, 1, 4, 22, 152, 1251, 11980, 130908, 1607488, ...
Examples:
A062817(4) = 3^1*1^3 + 2^2*2^2 + 1^3*3^1 = 22 ;
A062817(5) = 4^1*1^4 + 3^2*2^3 + 2^3*3^2 + 1^4*4^1 = 152 ;
A062817(6) = 5^1*1^5 + 4^2*2^4 + 3^3*3^3 + 2^4*4^2 + 1^5*5^1 = 1251 ;

This formula looks similar to a self-convolution ... and, surprisingly
(to me),
it is indeed a self-convolution of an integer sequence: A132608 (below).

Can anyone find another (simpler?) formula or g.f. for either A062817 or
A132608?

Also, what is the value of the limit:
Limit_{n->infinity} [A062817(n+1)/A062817(n)] / n = c

At n=12000, c ~ 1.35936744... and is decreasing.
Does the limit  c = exp(1)/2 ?

Thanks,
Paul
------------------------------------------
A132608

1,2,9,58,469,4530,50491,634790,8861043,135750454,2262315973,
40726646802,787471241647,16275700505510,358103286781293,
8357593147404346,206241859929682177,5366082228239257410

Self-convolution square-root of A062817 (offset 2);
thus g.f. A(x) satisfies:
A(x)^2 = Sum(n>=2} A062817(n)*x^n,
where A062817(n) = Sum_{k=0..n} (n-k)^k*k^(n-k).

EXAMPLE.
A(x) = x + 2x^2 + 9x^3 + 58x^4 + 469x^5 + 4530x^6 +...+ a(n)*x^n +...
A(x)^2 = x^2 + 4x^3 + 22x^4 + 152x^5 + 1251x^6 +...+ A062817(n)*x^n +...

(PARI)
{a(n)=polcoeff((sum(m=2,n+1,sum(k=0,m,(m-k)^k*k^(m-k))*x^m
+x*O(x^(n+1))))^(1/2),n)}
END.

Dear Seqfans,  I have about 900 old emails with suggestions
for sequences in them.  The earliest one was from Wouter,
and I have based two sequences on it, see below.
Could someone compute more terms?  This should be easy
with Magma, I think.
Neil

%I A131976
%S A131976 1,1,5
%N A131976 Let G be the full icosahedral group, of order 120.  Let v_1, ..., v_20 be the vertices of the dodecahedron.  Let S(n) be the set of vectors v_{i_1} + v_{i_2} + ... + v_{i_n} where 1 <= i_1 <= i_2 <= ... <= i_n <= 20.  Then a(n) = number of orbits of G on S(n).
%H A131976 Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/DodecahedralVectorSum.xls">Excel spreadsheet</a>
%e A131976 For 2 vertices, there are 5 different sets:
%e A131976 {10 pairs with norm^2 of sum = 0.000}
%e A131976 {30 pairs with 1.000}
%e A131976 {60, 2.618}
%e A131976 {60, 5.236}
%e A131976 {30, 6.854}
%e A131976 the norm^2 is taken with the side of the pentagons = 1.
%e A131976 And of course 10+30+60+60+30 = 190 = 20 choose 2
%O A131976 0,3
%K A131976 nonn,bref,more
%A A131976 njas, Oct 06 2007, based on an email message from Wouter Meeussen (wouter.meeussen(AT)pandora.be) on Dec 27 2004.

%I A131977
%S A131977 1,1,4
%N A131977 Analogue of A131976 for the icosahedron.
%H A131977 Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/DodecahedralVectorSum.xls">Excel spreadsheet</a>
%O A131977 0,3
%K A131977 nonn,bref,more
%A A131977 njas, Oct 06 2007, based on an email message from Wouter Meeussen (wouter.meeussen(AT)pandora.be) on Dec 27 2004.

```