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A319546 a(n) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16*17*18 - ... + (up to n).

%I A319546 #13 Oct 05 2018 08:00:03
%S A319546 1,2,6,24,120,720,713,664,216,-4320,-54720,-664560,-664547,-664378,
%T A319546 -661830,-620880,78000,12701520,12701501,12701140,12693540,12525960,
%U A319546 8663640,-84207600,-84207575,-84206950,-84190050,-83716200,-69957000,343310400,343310369
%N A319546 a(n) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16*17*18 - ... + (up to n).
%C A319546 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=6.
%C A319546 An alternating version of A319207.
%F A319546 a(n) = (-1)^floor(n/6) * Sum_{i=1..5} (1-sign((n-i) mod 6)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/6)+1) * (1-sign(i mod 6)) * (Product_{j=1..6} (i-j+1)).
%e A319546 a(1) = 1;
%e A319546 a(2) = 1*2 = 2;
%e A319546 a(3) = 1*2*3 = 6;
%e A319546 a(4) = 1*2*3*4 = 24;
%e A319546 a(5) = 1*2*3*4*5 = 120;
%e A319546 a(6) = 1*2*3*4*5*6 = 720;
%e A319546 a(7) = 1*2*3*4*5*6 - 7 = 713;
%e A319546 a(8) = 1*2*3*4*5*6 - 7*8 = 664;
%e A319546 a(9) = 1*2*3*4*5*6 - 7*8*9 = 216;
%e A319546 a(10) = 1*2*3*4*5*6 - 7*8*9*10 = -4320;
%e A319546 a(11) = 1*2*3*4*5*6 - 7*8*9*10*11 = -54720;
%e A319546 a(12) = 1*2*3*4*5*6 - 7*8*9*10*11*12 = -664560;
%e A319546 a(13) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13 = -664547;
%e A319546 a(14) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14 = -664378;
%e A319546 a(15) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15 = -661830;
%e A319546 a(16) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16 = -620880;
%e A319546 a(17) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16*17 = 78000; etc.
%t A319546 a[n_]:=(-1)^Floor[n/6]*Sum[(1-Sign[Mod[n-i,6]])*Product[n-j+1,{j,1,i}],{i,1,5}]+Sum[(-1)^(Floor[i/6]+1)*(1-Sign[Mod[i,6]])*Product[i-j+1,{j,1,5}],{i,1,n}]; Array[a, 30] (* _Stefano Spezia_, Sep 23 2018 *)
%Y A319546 For similar sequences, see: A001057 (k=1), A319373 (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), this sequence (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).
%Y A319546 Cf. A319207.
%K A319546 sign,easy
%O A319546 1,2
%A A319546 _Wesley Ivan Hurt_, Sep 22 2018