[seqfan] Re: Is A093468 identical to Partial sums of A001710 Order of alternating group A_n?
franktaw at netscape.net
franktaw at netscape.net
Thu May 13 01:53:58 CEST 2010
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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