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.

2, 1, 2, -1, 10, -43, 266, -1853, 14834, -133495, 1334962, -14684569, 176214842, -2290792931, 32071101050, -481066515733, 7697064251746, -130850092279663, 2355301661033954, -44750731559645105, 895014631192902122, -18795307255050944539

Offset: 1

Author

Zhi-Wei Sun, Jul 17 2010

Keywords

• sign

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

• Seiichi Manyama, Table of n, a(n) for n = 1..451
• 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]

Crossrefs

Cf. A000110, A000166.

Programs

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 *)

Formula

a(n) = a(n-1)*(1-n)+n+1. - Jon Maiga, Jul 10 2021