please doublecheck A094638

[franktaw at netscape.net]

I concur with (i) to (iii).

I have more problems with the description.

A. Shouldn't the range be 1 to p, instead of 1 to p-1?  This would 
apply both to the text "for i=1 to p-1" and the "i=1..p-1" in the sum.

B. "T(i,p) satisfies Sum_{i=1..p-1} T(i,p) * p^(p-1-i) / (p-1)!" seems 
to be incomplete.  I think it needs to say "T(i,p) satisfies Sum... = " 
something (maybe a Catalan number?).

Franklin T. Adams-Watters


-----Original Message-----
From: Emeric Deutsch <deutsch at duke.poly.edu>

Please doublecheck the following comments on A094638. 
Current A094638 reproduced below. 
 
(i) The "except" in the COMMENT does not make sense. Even those 
"exceptions" are Stirling numbers of 1st kind. More precisely, 
unsigned Stirling numbers of the first kind. 
 
(ii) Sequence starts with five 1's. It should start with four 1's. 
This explains why current Table is incorrect. Rows should be: 
1; 
1,1; 
1,3,2; 
1,6,11,6; 
1,10,35,50,24; 
1,15,85,225,274,120; (this is mentioned in the example). 
 
(iii) A formula is 
T(n,k)=abs(stirling1(n,n+1-k)). 
 
Neil: I will submit the corrected version as soon as this 
is doublechecked, preferably also by the author. 
 
Emeric 
 
------------------- 
 
A094638 Triangle read by rows giving coefficients of sum formulae of 
Catalan numbers. The p-th row (p>=1) contains T(i,p) for i=1 to p-1, 
where T(i,p) satisfies Sum_{i=1..p-1} T(i,p) * p^(p-1-i) / (p-1)!. 
 
 1, 1, 1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24, 1, 15, 85, 225, 
274, 120, 1, 21, 175, 735, 1624, 1764, 720, 1, 28, 322, 1960, 6769, 
13132, 13068, 5040, 1, 36, 546, 4536, 22449, 67284, 118124, 109584, 
40320, 1, 45, 870, 9450, 63273, 269325, 723680, 1172700 (list; table; 
graph) 
 
 OFFSET 1,6 
 
 COMMENT The coefficients T(i,p) of each row are indeed the Stirling 
numbers of 1st kind (see A008276, A000914, A001303, A000915, A053567) 
except for the second one which is the triangular number C(p-1,2) (see 
A000217) and the last one which is the factorial (p-2)! (see A000142). 
Further, the sum of coefficients on p-th row gives (p-1)!. 
 
 LINKS A. F. Labossiere, Sobalian Coefficients. 
 
A. F. Labossiere, Miscellaneous. 
 
 EXAMPLE Row 7 contains 1,15,85,225,274,120, so the 7th Catalan number 
equals the following : 
 
(1*7^5 +15*7^4 +85*7^3 +225*7^2 +274*7 +120)/6! = 132. 
 
 CROSSREFS Cf. A000108, A014137, A001246, A033536, A000984, A094639, 
A006134, A082894, A002897, A079727. 
 
Sequence in context: A111049 A088617 A008276 this_sequence A115755 
A016556 A067050 
 
Adjacent sequences: A094635 A094636 A094637 this_sequence A094639 
A094640 A094641 
 
 KEYWORD easy,nonn,tabl 
 
 AUTHOR Andre F. Labossiere (sobal(AT)laposte.net), May 17 2004