A331213 - OEIS

%I #29 Sep 08 2022 08:46:25

%S 1,0,1,-2,5,-4,13,-27,89,-80,191,-450,2365,-1182,3221,-13034,40433,

%T -22320,96373,-193761,772981,-728930,1599357,-3428425,21411337,

%U -13595724,31407273,-110011850,377746853,-198079308,1096983421,-2241234465,7565512161,-6472208192

%N a(n) = 1 + Sum_{i=1..n} (-1)^i * Product_{j=1..i} floor(n/j).

%C Compare to the exponential series: exp(-n) = 1 - n + n*(n/2) - n*(n/2)*(n/3) + n*(n/2)*(n/3)*(n/4) - ...

%H Seiichi Manyama, <a href="/A331213/b331213.txt">Table of n, a(n) for n = 0..1000</a>

%e a(4) = 1 - 4 + 4*floor(4/2) - 4*floor(4/2)*floor(4/3) + 4*floor(4/2)*floor(4/3)*floor(4/4) = 1 - 4 + 4*2 - 4*2*1 + 4*2*1*1 = 5.

%t a[n_] := 1 + Sum[(-1)^i * Product[Floor[n/j], {j, 1, i}],{i, 1, n}]; Array[a, 34, 0] (* _Amiram Eldar_, Jan 13 2020 *)

%o (PARI) {a(n) = 1+sum(i=1, n, (-1)^i*prod(j=1, i, floor(n/j)))}

%o (Magma) [1] cat [1+&+[(-1)^i*(&*[Floor(n/j):j in [1..i]]):i in [1..n]]:n in [1..33]]; // _Marius A. Burtea_, Jan 13 2020

%Y Cf. similar sequences: A075885 (b=1), A208060 (b=2).

%Y Cf. A010786.

%K sign

%O 0,4

%A _Seiichi Manyama_, Jan 12 2020