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

%I #10 Oct 05 2018 08:06:41

%S 8,56,336,1680,6720,20160,40320,40320,40304,40080,36960,-3360,-483840,

%T -5725440,-57617280,-518878080,-518878056,-518877528,-518865936,

%U -518623056,-513777600,-421968960,1225486080,29135312640,29135312608,29135311648,29135282880

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

%C For similar sequences that alternate in descending blocks of k natural numbers, we have: a(n) = (-1)^floor(n/k) * Sum_{j=1..k-1} (floor((n-j)/k) - floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (-1)^(floor(j/k)+1) * (floor(j/k) - floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=8.

%e a(1) = 8;

%e a(2) = 8*7 = 56;

%e a(3) = 8*7*6 = 336;

%e a(4) = 8*7*6*5 = 1680;

%e a(5) = 8*7*6*5*4 = 6720;

%e a(6) = 8*7*6*5*4*3 = 20160;

%e a(7) = 8*7*6*5*4*3*2 = 40320;

%e a(8) = 8*7*6*5*4*3*2*1 = 40320;

%e a(9) = 8*7*6*5*4*3*2*1 - 16 = 40304;

%e a(10) = 8*7*6*5*4*3*2*1 - 16*15 = 40080;

%e a(11) = 8*7*6*5*4*3*2*1 - 16*15*14 = 36960;

%e a(12) = 8*7*6*5*4*3*2*1 - 16*15*14*13 = -3360;

%e a(13) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12 = -483840;

%e a(14) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11 = -5725440;

%e a(15) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10 = -57617280;

%e a(16) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 = -518878080;

%e a(17) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24 = -518878056;

%e a(18) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24*23 = -518877528;

%e a(19) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24*23*22 = -518865936;

%e etc.

%p a:=(n,k)->(-1)^(floor(n/k))* add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add( (-1)^(floor(j/k)+1)*(floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,8),n=1..30); # _Muniru A Asiru_, Sep 30 2018

%Y For similar sequences, see: A001057 (k=1), A319885 (k=2), A319886 (k=3), A319887 (k=4), A319888 (k=5), A319889 (k=6), A319890 (k=7), this sequence (k=8), A319892 (k=9), A319893 (k=10).

%K sign,easy

%O 1,1

%A _Wesley Ivan Hurt_, Sep 30 2018