[seqfan] Re: Is A093468 identical to Partial sums of A001710 Order of alternating group A_n?
Jonathan Post
jvospost3 at gmail.com
Thu May 13 03:34:10 CEST 2010
Thank you, "franktaw", that is exactly right. You've helped me and
OEIS yet again.
It is a reason that I've been submitting (mixed in with other seqs)
these partial sums -- and have turned up 3 or more cases that connect
two existing seqs that don't crossreference each other in either
direction.
In my humble opinion, this is such an example, where each could xref
each other, your your proof as a comment.
Thanks again,
Jonathan Vos Post
On Wed, May 12, 2010 at 4:53 PM, <franktaw at netscape.net> wrote:
> Yes, they are the same numbers.
>
> Note, first of all, that there is an off-by-one problem: A093468 has
> offset 1, while A001710 has offset 0. I will use offset 0 here.
>
> For the sums of A001710, we have a(n) = sum(0 <= k <= n, (k! +
> [k<=1])/2), where by [k<=1] we mean 1 if k<=1, and 0 otherwise. (This
> is Knuth's notation.)
>
> Then for n > 1, a(n) + sum(0<=i<=n, a(n)-a(i)) = a(n) + sum(0<=i<k<=n,
> (k! + [k<=1])/2) = sum(0<k<=n, k*(k! + [k<=1])/2). But k*k! = (k+1)! -
> k!, so the terms in the sum telescope, and the sum is just (k+1)!/2.
> And this is exactly what we need it to be.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Jonathan Post <jvospost3 at gmail.com>
>
> Is
> A093468 a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum
> {a(n)-a(i), i = 1 to n}.
> identical to
> Partial sums of A001710 Order of alternating group A_n, or number of
> even permutations of n letters.
>
> Both begin: 1, 2, 3, 6, 18, 78, 438, 2958, 23118, ...
>
>
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