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%I #10 Oct 05 2018 08:06:47
%S 9,72,504,3024,15120,60480,181440,362880,362880,362862,362574,357984,
%T 289440,-665280,-13003200,-160030080,-1763959680,-17642862720,
%U -17642862693,-17642862018,-17642845170,-17642441520,-17633175120,-17429735520,-13167191520,71870561280
%N a(n) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13*12*11*10 + 27*26*25*24*23*22*21*20*19 - ... + (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=9.
%e a(1) = 9;
%e a(2) = 9*8 = 72;
%e a(3) = 9*8*7 = 504;
%e a(4) = 9*8*7*6 = 3024;
%e a(5) = 9*8*7*6*5 = 15120;
%e a(6) = 9*8*7*6*5*4 = 60480;
%e a(7) = 9*8*7*6*5*4*3 = 181440;
%e a(8) = 9*8*7*6*5*4*3*2 = 362880;
%e a(9) = 9*8*7*6*5*4*3*2*1 = 362880;
%e a(10) = 9*8*7*6*5*4*3*2*1 - 18 = 362862;
%e a(11) = 9*8*7*6*5*4*3*2*1 - 18*17 = 362574;
%e a(12) = 9*8*7*6*5*4*3*2*1 - 18*17*16 = 357984;
%e a(13) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15 = 289440;
%e a(14) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14 = -665280;
%e a(15) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13 = -13003200;
%e a(16) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13*12 = -160030080;
%e a(17) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13*12*11 = -1763959680;
%e a(18) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13*12*11*10 = -17642862720;
%e a(19) = 9*8*7*6*5*4*3*2*1 - 18*17*16*15*14*13*12*11*10 + 27 = -17642862693;
%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,9),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), this sequence (k=9), A319893 (k=10).
%K sign,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 30 2018
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