[seqfan] Re: Double Factorials

Henry Gould gould at math.wvu.edu
Sun Feb 1 21:16:10 CET 2009

Dear Maximilian,

Thanks for response. I shall comment on each item.

= = = = = = =

Maximilian Hasler wrote:
> Very legitimate question.
> Please let me know if you get the information in a private message.
> I can send you the PDF of this one:
> B. E. Meserve: Double Factorials
> The American Mathematical Monthly, Vol. 55, No. 7 (Aug. - Sep., 1948),
> pp. 425-426
This is one I knew about ever since it was published 60 years ago, but 
tells nothing about the origin of the notation.
> But it's certainly not the oldest.
> There is a citation from 1870 in the OEIS record
> http://www.research.att.com/~njas/sequences/A001147 :
> E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15
> (1870), 361-376.
> but I did not (yet) see the paper, so I'm not sure the !! notation had
> been used.
> Actually I see the comment
> %C A001147 The solution to Schroeder's third problem.
> so I don't think the notation was used.
Yes, I have seen the old journals, but doesn't help.
> However I found (via wikipedia)
> "... Arbogast who introduces between 1808 and 1816 the actual notation: n!."
> so it is not excluded that !! also was introduced before 1900.
I have a suspicion the double exclamatory mark was started in the first 
part of the 1900's. ????
Eric Weisstein, Wolfram, Wikipedia, etc, give no hint and it is not 
mentioned in Florian Cajori's History of Mathematical Notations (2 

The notation is also not used in NBS book (Abramowitz and Stegun), nor 
in CRC Handbook back as far as 1938, nor in Burington, Dwight, and other 
tables. I have searched in histories and hundreds of book with no 
result. So who or when it originated remains a puzzle to me.

---Henry Gould
> Regards,
> Maximilian
> On Thu, Jan 29, 2009 at 7:04 PM, Henry Gould <gould at math.wvu.edu> wrote:
>> Can anyone supply the name (and publication date_ of the originator(s)
>> of the double factorial n!!
>> The notation is usually used when giving the formulas for Wallis'
>> integral of powers of sine and cosine from 0 to pi. Also are there old
>> expository papers giving this information?
>> Henry Gould
>> _______________________________________________
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