please doublecheck A094638

[Emeric Deutsch]

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