login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A179508
a(n) is the unique integer such that Sum_{k=0..p-1} b(k)/(-n)^k == a(n) (mod p) for any prime p not dividing n, where b(0), b(1), b(2), ... are Bell numbers given by A000110.
3
2, 1, 2, -1, 10, -43, 266, -1853, 14834, -133495, 1334962, -14684569, 176214842, -2290792931, 32071101050, -481066515733, 7697064251746, -130850092279663, 2355301661033954, -44750731559645105, 895014631192902122, -18795307255050944539
OFFSET
1,1
COMMENTS
On July 17, 2010 Zhi-Wei Sun conjectured that a(n) exists for every n=1,2,3,... He noted that a(1)=2 since Sum_{k=0..p-1} (-1)^k * b(k) == b(p) (mod p), and conjectured that a(2)=1, a(3)=2, a(4)=-1, a(5)=10, a(6)=-43, a(7)=266, a(8)=-1853, a(9)=14834, a(10)=-133495. It seems that (-1)^(n-1)*a(n) > 0 for all n=3,4,5,...
I guess that a(2n) == (-1)^(n-1) (mod 4) and a(2n-1) == 2 (mod 4) for all n=1,2,3,... Perhaps a(2n-1) == 2 (mod 8) for every positive integer n. - Zhi-Wei Sun, Jul 18 2010
On August 5, 2010 Zhi-Wei Sun and Don Zagier proved that a(n) actually equals (-1)^(n-1)*D(n-1)+1, where D(0), D(1), D(2), ... are derangement numbers given by A000166. - Zhi-Wei Sun, Aug 07 2010
LINKS
Zhi-Wei Sun, Open Conjectures on Congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2010.
Zhi-Wei Sun, On Apery numbers and generalized central trinomial coefficients, preprint, arXiv:1006.2776 [math.NT], 2010-2011.
Zhi-Wei Sun, A conjecture on Bell numbers (a message to Number Theory List on July 17, 2010) [From Zhi-Wei Sun, Jul 18 2010]
Zhi-Wei Sun and Don Zagier, On a curious property of Bell numbers, Bulletin of the Australian Mathematical Society, Volume 84, Issue 1, August 2011. [Zhi-Wei Sun, Aug 07 2010]
FORMULA
a(n) = a(n-1)*(1-n)+n+1. - Jon Maiga, Jul 10 2021
MAPLE
A179508:= n-> (-1)^n*(n!*add((-1)^(k)/k!, k=0..n))+1 : seq(A179508(n), n=0..21);
# second program:
G(x):=(2-x)*exp(-x)/(1-x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1], x) od: x:=0; seq((-1)^n*f[n], n=0..21); # Mélika Tebni, Jul 10 2021
MATHEMATICA
a[1] = 2;
a[n_]:=a[n]=a[n-1]*(1-n)+n+1;
Array[a, 30] (* Jon Maiga, Jul 10 2021 *)
CROSSREFS
Sequence in context: A248516 A097749 A126906 * A134304 A211096 A134569
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Jul 17 2010
STATUS
approved