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%I #10 Oct 05 2018 08:06:54
%S 10,90,720,5040,30240,151200,604800,1814400,3628800,3628800,3628780,
%T 3628420,3621960,3512520,1768320,-24278400,-387072000,-5075481600,
%U -60945696000,-670438944000,-670438943970,-670438943130,-670438919640,-670438286280,-670421843280
%N a(n) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16*15*14*13*12*11 + 30*19*18*17*16*15*14*13*12*11 - ... + (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=10.
%e a(1) = 10;
%e a(2) = 10*9 = 90;
%e a(3) = 10*9*8 = 720;
%e a(4) = 10*9*8*7 = 5040;
%e a(5) = 10*9*8*7*6 = 30240;
%e a(6) = 10*9*8*7*6*5 = 151200;
%e a(7) = 10*9*8*7*6*5*4 = 604800;
%e a(8) = 10*9*8*7*6*5*4*3 = 1814400;
%e a(9) = 10*9*8*7*6*5*4*3*2 = 3628800;
%e a(10) = 10*9*8*7*6*5*4*3*2*1 = 3628800;
%e a(11) = 10*9*8*7*6*5*4*3*2*1 - 20 = 3628780;
%e a(12) = 10*9*8*7*6*5*4*3*2*1 - 20*19 = 3628420;
%e a(13) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18 = 3621960;
%e a(14) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17 = 3512520;
%e a(15) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16 = 1768320;
%e a(16) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16*15 = -24278400;
%e a(17) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16*15*14 = -387072000;
%e a(18) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16*15*14*13 = -5075481600;
%e a(19) = 10*9*8*7*6*5*4*3*2*1 - 20*19*18*17*16*15*14*13*12 = -60945696000;
%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,10),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), A319891 (k=8), A319892 (k=9), this sequence (k=10).
%K sign,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 30 2018
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