A319549 a(n) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15*16 + 17*18*19*20*21*22*23*24 - ... + (up to n).

1, 2, 6, 24, 120, 720, 5040, 40320, 40311, 40230, 39330, 28440, -114120, -2121840, -32392080, -518878080, -518878063, -518877774, -518872266, -518761800, -516436200, -465156720, 716713200, 29135312640, 29135312615, 29135311990, 29135295090, 29134821240

Offset: 1

Author

Wesley Ivan Hurt, Sep 22 2018

Keywords

• sign
• easy

Comments

In general, for alternating sequences that multiply the first k natural numbers, and subtract/add the products of the next k natural numbers (preserving the order of operations up to n), we have a(n) = (-1)^floor(n/k) * Sum_{i=1..k-1} (1-sign((n-i) mod k)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/k)+1) * (1-sign(i mod k)) * (Product_{j=1..k} (i-j+1)). Here k=8.

An alternating version of A319209.

Examples

			a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6*7 = 5040;
a(8) = 1*2*3*4*5*6*7*8 = 40320;
a(9) = 1*2*3*4*5*6*7*8 - 9 = 40311;
a(10) = 1*2*3*4*5*6*7*8 - 9*10 = 40230;
a(11) = 1*2*3*4*5*6*7*8 - 9*10*11 = 39330;
a(12) = 1*2*3*4*5*6*7*8 - 9*10*11*12 = 28440;
a(13) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13 = -114120;
a(14) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14 = -2121840;
a(15) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15 = -32392080;
a(16) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15*16 = -518878080;
a(17) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15*16 + 17 = -518878063; etc.
		

Crossrefs

For similar sequences, see: A001057 (k=1), A319373 (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), A319546 (k=6), A319547 (k=7), this sequence (k=8), A319550 (k=9), A319551 (k=10).

Cf. A319209.

Programs

Mathematica

a[n_]:=(-1)^Floor[n/8]*Sum[(1-Sign[Mod[n-i,8]])*Product[n-j+1,{j,1,i}],{i,1,7}]+Sum[(-1)^(Floor[i/8]+1)*(1-Sign[Mod[i,8]])*Product[i-j+1,{j,1,7}],{i,1,n}]; Array[a, 30] (* Stefano Spezia, Sep 23 2018 *)
Table[Total[Times@@@Partition[Riffle[Times@@@Partition[Range[n],UpTo[8]],{1,-1},{1,-1,2}],2]],{n,30}] (* Harvey P. Dale, Oct 05 2024 *)

Formula

a(n) = (-1)^floor(n/8) * Sum_{i=1..7} (1-sign((n-i) mod 8)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/8)+1) * (1-sign(i mod 8)) * (Product_{j=1..8} (i-j+1)).