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A319551 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*28*29*30 - ... + (up to n).

%I A319551 #16 Oct 05 2018 08:02:28
%S A319551 1,2,6,24,120,720,5040,40320,362880,3628800,3628789,3628668,3627084,
%T A319551 3604776,3268440,-2136960,-94389120,-1760693760,-33518499840,
%U A319551 -670438944000,-670438943979,-670438943538,-670438933374,-670438688976,-670432568400,-670273178400
%N A319551 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*28*29*30 - ... + (up to n).
%C A319551 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=10.
%C A319551 An alternating version of A319212.
%F A319551 a(n) = (-1)^floor(n/10) * Sum_{i=1..9} (1-sign((n-i) mod 10)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/10)+1) * (1-sign(i mod 10)) * (Product_{j=1..10} (i-j+1)).
%e A319551 a(1) = 1;
%e A319551 a(2) = 1*2 = 2;
%e A319551 a(3) = 1*2*3 = 6;
%e A319551 a(4) = 1*2*3*4 = 24;
%e A319551 a(5) = 1*2*3*4*5 = 120;
%e A319551 a(6) = 1*2*3*4*5*6 = 720;
%e A319551 a(7) = 1*2*3*4*5*6*7 = 5040;
%e A319551 a(8) = 1*2*3*4*5*6*7*8 = 40320;
%e A319551 a(9) = 1*2*3*4*5*6*7*8*9 = 362880;
%e A319551 a(10) = 1*2*3*4*5*6*7*8*9*10 = 3628800;
%e A319551 a(11) = 1*2*3*4*5*6*7*8*9*10 - 11 = 3628789;
%e A319551 a(12) = 1*2*3*4*5*6*7*8*9*10 - 11*12 = 3628668;
%e A319551 a(13) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13 = 3627084;
%e A319551 a(14) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14 = 3604776;
%e A319551 a(15) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15 = 3268440;
%e A319551 a(16) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16 = -2136960;
%e A319551 a(17) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17 = -94389120;
%e A319551 a(18) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17*18 = -1760693760;
%e A319551 a(19) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17*18*19 = -33518499840; etc.
%t A319551 a[n_]:=(-1)^Floor[n/10]*Sum[(1-Sign[Mod[n-i,10]])*Product[n-j+1,{j,1,i}],{i,1,9}]+Sum[(-1)^(Floor[i/10]+1)*(1-Sign[Mod[i,10]])*Product[i-j+1,{j,1,10}],{i,1,n}]; Array[a, 30] (* _Stefano Spezia_, Sep 23 2018 *)
%Y A319551 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), A319550 (k=9), this sequence (k=10).
%Y A319551 Cf. A319212.
%K A319551 sign,easy
%O A319551 1,2
%A A319551 _Wesley Ivan Hurt_, Sep 22 2018