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A319550 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*25*26*27 - ... + (up to n).

%I A319550 #16 Oct 05 2024 14:31:34
%S A319550 1,2,6,24,120,720,5040,40320,362880,362870,362770,361560,345720,
%T A319550 122640,-3240720,-57294720,-979816320,-17642862720,-17642862701,
%U A319550 -17642862340,-17642854740,-17642687160,-17638824840,-17545953600,-15220134720,45348065280,1683112193280
%N A319550 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*25*26*27 - ... + (up to n).
%C A319550 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=9.
%C A319550 An alternating version of A319211.
%H A319550 Harvey P. Dale, <a href="/A319550/b319550.txt">Table of n, a(n) for n = 1..1000</a>
%F A319550 a(n) = (-1)^floor(n/9) * Sum_{i=1..8} (1-sign((n-i) mod 9)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/9)+1) * (1-sign(i mod 9)) * (Product_{j=1..9} (i-j+1)).
%e A319550 a(1) = 1;
%e A319550 a(2) = 1*2 = 2;
%e A319550 a(3) = 1*2*3 = 6;
%e A319550 a(4) = 1*2*3*4 = 24;
%e A319550 a(5) = 1*2*3*4*5 = 120;
%e A319550 a(6) = 1*2*3*4*5*6 = 720;
%e A319550 a(7) = 1*2*3*4*5*6*7 = 5040;
%e A319550 a(8) = 1*2*3*4*5*6*7*8 = 40320;
%e A319550 a(9) = 1*2*3*4*5*6*7*8*9 = 362880;
%e A319550 a(10) = 1*2*3*4*5*6*7*8*9 - 10 = 362870;
%e A319550 a(11) = 1*2*3*4*5*6*7*8*9 - 10*11 = 362770;
%e A319550 a(12) = 1*2*3*4*5*6*7*8*9 - 10*11*12 = 361560;
%e A319550 a(13) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13 = 345720;
%e A319550 a(14) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14 = 122640;
%e A319550 a(15) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15 = -3240720;
%e A319550 a(16) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16 = -57294720;
%e A319550 a(17) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17 = -979816320;
%e A319550 a(18) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17*18 = -17642862720;
%e A319550 a(19) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17*18 + 19 = -17642862701; etc.
%t A319550 Table[Total[Times@@@Partition[Riffle[Times@@@Partition[Range[n],UpTo[9]],{1,-1},{1,-1,2}],2]],{n,30}] (* _Harvey P. Dale_, Oct 05 2024 *)
%Y A319550 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), A319549 (k=8), this sequence (k=9), A319551 (k=10).
%Y A319550 Cf. A319211.
%K A319550 sign,easy
%O A319550 1,2
%A A319550 _Wesley Ivan Hurt_, Sep 22 2018