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.

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.