%I #13 Oct 05 2018 07:58:50
%S 1,2,6,24,120,720,5040,5032,4968,4320,-2880,-90000,-1230480,-17292240,
%T -17292225,-17292000,-17288160,-17218800,-15896880,10614960,568758960,
%U 568758938,568758454,568746816,568455360,560865360,355631760,-5398802640,-5398802611
%N a(n) = 1*2*3*4*5*6*7 - 8*9*10*11*12*13*14 + 15*16*17*18*19*20*21 - ... + (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=7.
%C An alternating version of A319208.
%F a(n) = (-1)^floor(n/7) * Sum_{i=1..6} (1-sign((n-i) mod 7)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/7)+1) * (1-sign(i mod 7)) * (Product_{j=1..7} (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 = 5032;
%e a(9) = 1*2*3*4*5*6*7 - 8*9 = 4968;
%e a(10) = 1*2*3*4*5*6*7 - 8*9*10 = 4320;
%e a(11) = 1*2*3*4*5*6*7 - 8*9*10*11 = -2880;
%e a(12) = 1*2*3*4*5*6*7 - 8*9*10*11*12 = -90000;
%e a(13) = 1*2*3*4*5*6*7 - 8*9*10*11*12*13 = -1230480;
%e a(14) = 1*2*3*4*5*6*7 - 8*9*10*11*12*13*14 = -17292240;
%e a(15) = 1*2*3*4*5*6*7 - 8*9*10*11*12*13*14 + 15 = -17292225;
%e a(16) = 1*2*3*4*5*6*7 - 8*9*10*11*12*13*14 + 15*16 = -17292000; etc.
%t a[n_]:=(-1)^Floor[n/7]*Sum[(1-Sign[Mod[n-i,7]])*Product[n-j+1,{j,1,i}],{i,1,6}]+Sum[(-1)^(Floor[i/7]+1)*(1-Sign[Mod[i,7]])*Product[i-j+1,{j,1,6}],{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), this sequence (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).
%Y Cf. A319208.
%K sign,easy
%O 1,2
%A _Wesley Ivan Hurt_, Sep 22 2018