Triangle

Tue Apr 15 19:58:03 CEST 2008

Am 15.04.2008 18:09 schrieb Gottfried Helms:

> So
>   U_t(x,h) = a1(t,h)*x/1! + a2(t,h)*x^2/2! + a3(t,h)*x^3/3! + ...
> 
> The coeffient-functions a_k() can be expressed as matrices, row-
> resp. column-scaled by t resp t^h.
Correction:
... row resp column-summed by ...

Gottfried Helms

gh> From seqfan-owner at ext.jussieu.fr  Tue Apr 15 15:33:24 2008
gh> Date: Tue, 15 Apr 2008 14:58:26 +0200
gh> From: Gottfried Helms <helms at uni-kassel.de>
gh> To: Seqfan at ext.jussieu.fr
gh> Subject: Triangle
gh> 
gh> Did someone come across this triangle?
gh> 
gh>                           1
gh>                      1    1    1
gh>                 1    3    4    3    1
gh>            1    7   13   19   13    6   1
gh>       1   15   40   85   96   75   35  10  1
gh>   1  31  121  335  560  616  471  240  80  15  1
gh>  ...
gh> 
gh> 
gh> (not found in OEIS)
gh> 
gh> Apparently related to Stirling numbers (1'st or 2'nd kind)
gh>        1
gh>      1   1
gh>    1   3   1
gh>  1   7   6   1
gh>  ...
gh> 
gh> Gottfried Helms

After noticing the sum formula in A000258 and further
decomposition of the Bell numbers in there in terms of S2
(Stirling numbers of the second kind) one can summarize
the "Helms" array  H(n,j),  rows n=1,2,3,4,...,
terms j=2,3,4,..,2n enumerated from the left to the right as:

H(n,j)= sum_{k+l=j) X(k,l,n)

where the sum is over the diagonal of an auxiliary upper right
triangle defined as
X(k,l,n) := S2(n,k)*S2(k,l) for 1<=k<=n and 1<=l<=k.

where H(.,.) becomes

1 
1 1 1 
1 3 4 3 1 
1 7 13 19 13 6 1 
1 15 40 85 96 75 35 10 1 
1 31 121 335 560 616 471 240 80 15 1 
1 63 364 1253 2891 4221 4502 3353 1806 665 161 21 1 
1 127 1093 4599 13923 26222 36225 36205 26895 14756 5887 1638 294 28 1 
1 255 3280 16845 64366 153531 264033 336792 322576 236421 131587 55272 16989 3654 498 36 1 
1 511 9841 62095 290590 865332 1810747 2850870 3391455 3136381 2258413 1269960 552280 182595 44367 7500 795 45 1 
...

Maple Implementation:

X := proc(k,l,n)
end:

H := proc(n,j)
end:

for n from 1 to 10 do
od:

dww> From seqfan-owner at ext.jussieu.fr  Mon Apr 14 15:13:15 2008
dww> From: "David W. Wilson" <wilson.d at anseri.com>
dww> To: <seqfan at ext.jussieu.fr>
dww> Subject: RE: Sum of prime rereciprocals?
dww> Date: Mon, 14 Apr 2008 09:08:31 -0400
dww> 
dww> Why not just sum of reciprocals, e.g:
dww> 
dww> a(n) = least k with sum(j = n..k; 1/j) >= 1.
dww> 
dww> It's an interesting sequence, which starts
dww> 
dww>    1,4,7,10,12,15,...
dww> 
dww> If you compare it to [en] = A022843
dww> 
dww>    2,5,8,10,13,16,...
dww> 
dww> it appears that [en]-a(n) =
dww> 
dww>    1,1,1,0,1,1,1,1,1,1,0,...
dww> 
dww> consists of 0's and 1's.
dww> 
dww> The places where the zeroes occur are
dww> 
dww>     4, 11, 18, 25, 32, 36, 43, 50, 57, 64, 71, ...
dww> 
dww> whose differences always seem to be 4, 7 or 11.

There are some rather sharp estimates on this type of differences
between harmonic numbers in Theorem 3.2 of A. Sintamarian,
"A generalization of Euler's constant", Numer. Algor. 46 (2007) p 141-151
http://dx.doi.org/10.1007/s11075-007-9132-0
which may help to uncover such a pattern.

--
Richard

------ Original Message ------

> Let a(n) be the number of rings of order p^2, where p is the nth prime.
> 
> These are the 4th, 9th, 25th, 49th, etc. terms of A027623.
> 
> The sequence begins: 11,11,11,11
> 
> And continues?
> 
> Interestingly, the known terms of A027623 are only equal to 11 when n is
> the square of a prime, so perhaps both the conjecture and its converse
> are true.
> 
> A027623:
> 1, 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22,
> 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2, ? (>18590), 4, 4, 4, 121, 2, 4, 4,
> 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, 4, 22, 2, 118, 4, 104, 4, 4, 2,
> 44, 2, 4, 22, ? (> 829826)
> 
> Neil
> 
This was discussed in detail on math-fun back in June 1998.

The short answer is yes, both conjecture and its converse are true.
Unfortunately the link to C. Noebauer's paper on small rings seems to be
broken. I have a copy of it though. It's a short paper citing some theorems
about small rings. It only cites; it does not prove. In particular are the
results that:

A027623(p^2)=11, 

A027623(p^2)=3p+50 for p an odd prime (52 if p=2)

A037291(p^3)=12 for p an odd prime (11 if p=2)

Since A027623 is multiplicative, it's easy to see that the only possibility
for a(n)=11 is n=p^2 as a(p)=2, a(p^(k+1))>a(p^k).

John Conway did a proof of a(p^2)=11 in that discussion from 1998. I can post
it if anyone is interested.

Christian