A025016: 10-adic Sum of Factorials

[franktaw at netscape.net]

B(1) = Sum_{n>=0} n*n! = Sum_{n>=0} (n+1)!-n! = (Sum_{n>=1} n!) - 
(Sum_{n>=0} n!) = -1! = -1. 

***Correction - that should be ... = -0! = -1.
 
Generally, n^k * n! can be written as a linear function of n!, ..., 
(n+k-1)!; I suspect this will lead to your general relation, but I 
don't have time to dig into it now. 
 
Franklin T. Adams-Watters 
 
-----Original Message----- 
From: Paul D. Hanna <pauldhanna at juno.com> 
 
Seqfans, 
     Consider David Wilson's nice 10-adic constant A025016: 
x = Sum_{n>=0} n! (10-adic) 
=...92247479478684738621107994804323593105039052556442336528920420940314 

    
Investigating the Bell number analogue: 
  
   B(k) = Sum_{n>=0} n^k*n! (10-adic) 
  
I was quite surprised to find that 
  
   B(1) = -1 = ...99999999999 (10-adic). 
  
I went further to find the remarkable relation 
(see examples at bottom of message): 
  
   B(n) = A014182(n)*B(0) + A014619(n) 
  
But, wait - this is a base-independent formula !  
 
Does the same formula hold for p-adic bases other than 10-adic? I have 
not had time to check, but I think it does. 
  
This constant A025016 has a b-adic expression for all base b. 
I believe that the constant A025016 is base-independent, 
 
but the digits recorded in A025016 are base-10. 
    
Any comments? 
Thanks, 
       Paul 
  
Conjectured Formula: 
B(n) = A014182(n)*B(0) + A014619(n) 
Examples: 
B(1) = 0*B(0) - 1 
B(2) = -1*B(0) + 1 
B(3) =  1*B(0) + 1 
B(4) =  2*B(0) - 5 
B(5) = -9*B(0) + 5 
B(6) = 9*B(0) + 21 
B(7) = 50*B(0) - 105 
B(8) = -267*B(0) + 141 
B(9) = 413*B(0) + 777 
B(10) = 2180*B(0) - 5513 
B(11) = -17731*B(0) + 13209 
B(12) = 50533*B(0) + 39821 
B(13) = 110176*B(0) - 527525 
B(14) = -1966797*B(0) + 2257425 
B(15) = 9938669*B(0) - 41511 
  
  
A014182   
Expansion of exp(1-x-exp(-x)).  
1, 0, -1, 1, 2, -9, 9, 50, -267, 413, 2180, -17731, 50533, 110176, 
-1966797, 9938669, -8638718, -278475061, 2540956509, -9816860358,  
  
A014619  
Exponential generating function is -f(x) * int(exp(exp(-t)-1),t,0,x) 
where f(x) = exp(1-x-exp(-x)) is an exponential generating function for 
A014182.  
-1, 1, 1, -5, 5, 21, -105, 141, 777, -5513, 13209, 39821, -527525, 
2257425, 
  
END.