Error in A115866

[David W. Cantrell]

On Sunday, March 04, 2007, "Peter Pein" <petsie at dordos.net> wrote:
> Using
> paths[d_, n_] := Sum[Binomial[m, n]^d/2^(m + 1), {m, 0, ∞}]
> (d = dimension, n stands for the point (n,n,...,n)).
>
> I can not see any anomalies (at least for 1<= d,n <= 10)
>
> As the calculation for these values ran for ~15 minutes, I attach a
> csv-file for the impatient to experiment with.

I had _hoped_ to give you a substantial time-saver.

On my computer, using Mathematica,

Table[
Sum[Binomial[m, n]^d/2^(m + 1), {m, n, Infinity]}],
{d, 1, 4}, {n, 1, 4}]

took more than 5 seconds, while the same ouput was produced in less
than 0.05 seconds using a closed form

Table[
2^(-1 - n) HypergeometricPFQ[Table[1 + n, {d}], Table[1, {d - 1}],
1/2], {d, 1, 4}, {n, 1, 4}].

But as the common max of d and n increased, the advantage of my closed
form got less and less until, for 1<= d,n <= 7, both methods took
about 56 seconds. Going beyond that point, surely my closed-form
method would take _substantially longer_. Why?! Maybe it's due to a
slow implementation of hypergeometric functions in Mathematica when
there are many parameters, or maybe such evaluations are inherently 
slow...

Best regards,
David







Maybe it drops a word for brevity, and the more precise phrase would be


??  I don't have time to check this right now, but it *seems* plausible.


Yes, I suppose that's reasonable.  Originally I conceived of these operations
in terms of a general base B, and then took B=10 as an "arbitrary" example for
purposes of illustration.

For B=2 dismal add and multiply are just a GF2 "numbral" arithmetic 
you get with
bitwise binary operations.  For other "small" B they tend to have 
other special
as well as
friendlier to us decadacts doing hand calculations...






New version:

%I A088475
%S A088475 10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,
%T A088475 32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A088475 54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75
%N A088475 Numbers n such that dismal sum of the dismal prime divisors of n is >= n.
%e A088475 The only dismal prime that divides 10 is 90: 90*1 = 10 (cf. A087061, A087062, A08\
7097), and 90+1 = 91 >= 10, so 10 is a member. - njas, Mar 04 2007
%H A088475 D. Applegate, <a href="http://www.research.att.com/~njas/sequences/a087061.txt">C\
%H A088475 <a href="http://www.research.att.com/~njas/sequences/Sindx_Di.html#dismal">Index \
entries for sequences related to dismal arithmetic</a>
%Y A088475 Sequence in context: A063671 A123895 A100830 this_sequence A001637 A102494 A11788\
4
%Y A088475 Adjacent sequences: A088472 A088473 A088474 this_sequence A088476 A088477 A088478
%K A088475 nonn,base
%O A088475 1,1
%A A088475 David Applegate (david(AT)research.att.com), Nov 11 2003
%E A088475 Definition made more precise by Marc LeBrun, Mar 04 2007

Neil