# A062817: Self-Convolution Square-Root: Coincidental Observation

Paul D. Hanna pauldhanna at juno.com
Thu Oct 11 06:16:04 CEST 2007

```Seqfans,
Here is an accidental observation somewhat related to my question.
Consider Vladeta Jovovic's nice sequence A060435, where:
A060435(n) = number of functions f:{1,2,...,n}->{1,2,...,n} with even
cycles only.

It has the rather complicated formula for the n-th term:

a(n)=(n-1)!*Sum_{k=0..floor((n-2)/2)} (k+1)/2^(2*k+1)*binomial(2*k+2,
k+1)*n^(n-2-2*k)/(n-2-2*k)!.

Now if we square of the e.g.f. of A060435 after we include the terms
A060435(0)=1, A060435(1)=0,
then we find that the NEW sequence (A134095) has a simple formula:

a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).

and the e.g.f. follows from Vladeta's formula for A060435:

E.g.f.: A(x) = 1/(1 - LambertW(-x)^2 ).

Below I supply the initial terms of A134095 and the PARI code.

Just one of many accidental coincidences of the OEIS.
Paul

A134095 begins:
1,0,2,12,120,1480,22320,396564,8118656,188185680,4871980800,
139342178140,4363291266048,148470651659928,5455056815237120,
215238256785814500,9077047768435752960,407449611073696325536,

(PARI)
a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))
/* Generated by E.G.F. 1/(1 - LambertW(-x)^2 ): */
{a(n)=local(LambertW=-x*sum(k=0,n,(-x)^k*(k+1)^(k-1)/k!) +x*O(x^n));
n!*polcoeff(1/(1-subst(LambertW,x,-x)^2),n)}

END.
On Sat, 6 Oct 2007 01:38:29 -0400 Paul D. Hanna <pauldhanna at juno.com>
writes:
> 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)}
>
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