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%I #16 May 31 2022 15:40:46
%S 5,20,60,120,120,110,30,-600,-4920,-30120,-30105,-29910,-27390,2640,
%T 330240,330220,329860,323400,213960,-1530240,-1530215,-1529640,
%U -1516440,-1226640,4845360,4845330,4844490,4821000,4187640,-12255360,-12255325,-12254170,-12216090
%N a(n) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 - 20*19*18*17*16 + ... - (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=5.
%e a(1) = 5;
%e a(2) = 5*4 = 20;
%e a(3) = 5*4*3 = 60;
%e a(4) = 5*4*3*2 = 120;
%e a(5) = 5*4*3*2*1 = 120;
%e a(6) = 5*4*3*2*1 - 10 = 110;
%e a(7) = 5*4*3*2*1 - 10*9 = 30;
%e a(8) = 5*4*3*2*1 - 10*9*8 = -600;
%e a(9) = 5*4*3*2*1 - 10*9*8*7 = -4920;
%e a(10) = 5*4*3*2*1 - 10*9*8*7*6 = -30120;
%e a(11) = 5*4*3*2*1 - 10*9*8*7*6 + 15 = -30105;
%e a(12) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14 = -29910;
%e a(13) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13 = -27390;
%e a(14) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12 = 2640;
%e a(15) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 = 330240;
%e a(16) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 - 20 = 330220;
%e a(17) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 - 20*19 = 329860;
%e a(18) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 - 20*19*18 = 323400;
%e a(19) = 5*4*3*2*1 - 10*9*8*7*6 + 15*14*13*12*11 - 20*19*18*17 = 213960;
%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,5),n=1..40); # _Muniru A Asiru_, Sep 30 2018
%Y For similar sequences, see: A001057 (k=1), A319885 (k=2), A319886 (k=3), A319887 (k=4), this sequence (k=5), A319889 (k=6), A319890 (k=7), A319891 (k=8), A319892 (k=9), A319893 (k=10).
%K sign,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 30 2018
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