Matching first n signs of differences between consecutive terms

[Leroy Quet]

--- Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:

> I just submitted this sequence.
> 
> %I A137947
> %S A137947 3,13,13,13,13,13,13,13
> %N A137947 a(n) = k: k is smallest integer > 1
> such that sign(d(1)-d(2)) = sign(d(k)-d(k+1)),
> sign(d(2)-d(3)) =
> sign(d(k+1)-d(k+2)),...,sign(d(n-1)-d(n)) =
> sign(d(k+n-1)-d(k+n)), where sign is (-,0,+)
> and
> d(m) = the number of positive divisors of m.
> %Y A137947 A000005
> %O A137947 1
> %K A137947 ,more,nonn,
> %A A137947 Leroy Quet (qq-quet at mindspring.com),
> Feb 24 2008
> 

[rest snipped]

That "sign(d(n-1)-d(n))" in the N-line should
instead be
"sign(d(n)-d(n+1))".

(How did that pass the censors? I double-checked
it before sending it in.)

N-line again as it should be:


%N A137947 a(n) = k: k is smallest integer > 1
such that sign(d(1)-d(2)) = sign(d(k)-d(k+1)),
sign(d(2)-d(3)) = sign(d(k+1)-d(k+2)),...,
sign(d(n)-d(n+1)) = sign(d(k+n-1)-d(k+n)), where
sign is (-,0,+), and d(m) = the number of
positive divisors of m.

Neil: Please make appropriate correction before
you include this sequence in the database.


Thanks, sorry,
Leroy Quet




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Dear Seqfans and Associate OEIS Editors:

Here is the current entry for A106239:

%I A106239
%S A106239 0,0,0,1,0,0,15,0,0,0,222,0,0,0,0,3660,10,0,0,0,0,68295,525,0,0,0,0,0,
%T A106239 1436568,20307,0,0,0,0,0,0,33779340,727020,280,0,0,0,0,0,0,880107840,
%U A106239 25934184,31500,0,0,0,0,0,0,0,25201854045,950478210,2325015,0,0,0,0,0,0
%N A106239 Triangle read by rows: T(n,m) = number of graphs on n labeled nodes, with m components of size = order. Also number of graphs on n labeled nodes with m unicyclic components.
%F A106239 E.g.f.: exp(-y/2*ln(1+LambertW(-x))+y/2*LambertW(-x)-y/4*LambertW(-x)^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 04 2005
%F A106239 T(n, m)= sum N/D over the partitions of n:1K1+2K2+ ... +nKn, = with exactly m parts greater than 2, where N = n!*product_{1=<i<=n}= A057500(i)^Ki, and D = product_{1=<i<=n}(Ki!(i!)^Ki).
%e A106239 T(6,2)=10 because there are 10 such graphs of order 6 with 2 components. The value of T(n,m) is zero if and only if m > floor(n/3).
%Y A106239 Cf. A057500 and A106238 (similar formulae that can be used in the unlabeled case.).
%K A106239 nonn,tabl
%O A106239 1,7
%A A106239 Washington Bomfim (webonfim(AT)bol.com.br), May 03 2005

There are three other mentions of it:

%C A137916 The first values are row sums of A106239.
%Y A106238 Cf. A057276, A035512, A106237, A106239.
%H A106240 Washington Bomfim, <a href="http://webonfim.vilabol.uol.com.br/graphsunicycliccomponents.html">Illustration of A106239</a>

The other day Tom Zaslavsky, zaslav at math.binghamton.edu, wrote to me saying:

%N A106239 T(n,m) = # of simple graphs on n labeled nodes with m components, each component being unicyclic (one cycle).
%C A106239 I'm concerned about this sequence.
The largest possible number of components is floor(n/3), so T(5,2) = 0, but the table shows 10. 
The nonzero lower boundary should have slope -1/3, but its slope is -1/2.
Am I being completely stupid, or is there something wrong?
I'm interested because I'm editing Wikipedia on "pseudoforests" and \sum_m T(n,m) (which is apparently not in the database) should be the number of simple pseudoforests on n labelled vertices.

Can someone explain what this sequence is showing? Vladeta?
Thanks!
Neil