|
%I #13 Oct 05 2018 08:02:05
%S 1,2,6,24,120,720,5040,40320,40311,40230,39330,28440,-114120,-2121840,
%T -32392080,-518878080,-518878063,-518877774,-518872266,-518761800,
%U -516436200,-465156720,716713200,29135312640,29135312615,29135311990,29135295090,29134821240
%N 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).
%C 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.
%C An alternating version of A319209.
%F 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)).
%e a(1) = 1;
%e a(2) = 1*2 = 2;
%e a(3) = 1*2*3 = 6;
%e a(4) = 1*2*3*4 = 24;
%e a(5) = 1*2*3*4*5 = 120;
%e a(6) = 1*2*3*4*5*6 = 720;
%e a(7) = 1*2*3*4*5*6*7 = 5040;
%e a(8) = 1*2*3*4*5*6*7*8 = 40320;
%e a(9) = 1*2*3*4*5*6*7*8 - 9 = 40311;
%e a(10) = 1*2*3*4*5*6*7*8 - 9*10 = 40230;
%e a(11) = 1*2*3*4*5*6*7*8 - 9*10*11 = 39330;
%e a(12) = 1*2*3*4*5*6*7*8 - 9*10*11*12 = 28440;
%e a(13) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13 = -114120;
%e a(14) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14 = -2121840;
%e a(15) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15 = -32392080;
%e a(16) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15*16 = -518878080;
%e a(17) = 1*2*3*4*5*6*7*8 - 9*10*11*12*13*14*15*16 + 17 = -518878063; etc.
%t 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 *)
%Y 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).
%Y Cf. A319209.
%K sign,easy
%O 1,2
%A _Wesley Ivan Hurt_, Sep 22 2018
| 