1,2

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.

An alternating version of A319212.

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)).

a(1) = 1;

a(2) = 1*2 = 2;

a(3) = 1*2*3 = 6;

a(4) = 1*2*3*4 = 24;

a(5) = 1*2*3*4*5 = 120;

a(6) = 1*2*3*4*5*6 = 720;

a(7) = 1*2*3*4*5*6*7 = 5040;

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

a(9) = 1*2*3*4*5*6*7*8*9 = 362880;

a(10) = 1*2*3*4*5*6*7*8*9*10 = 3628800;

a(11) = 1*2*3*4*5*6*7*8*9*10 - 11 = 3628789;

a(12) = 1*2*3*4*5*6*7*8*9*10 - 11*12 = 3628668;

a(13) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13 = 3627084;

a(14) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14 = 3604776;

a(15) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15 = 3268440;

a(16) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16 = -2136960;

a(17) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17 = -94389120;

a(18) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17*18 = -1760693760;

a(19) = 1*2*3*4*5*6*7*8*9*10 - 11*12*13*14*15*16*17*18*19 = -33518499840; etc.

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 *)

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).

Cf. A319212.

Sequence in context: A360462 A072133 A230232 * A232985 A319212 A276844

Adjacent sequences: A319548 A319549 A319550 * A319552 A319553 A319554

sign,easy

Wesley Ivan Hurt, Sep 22 2018

approved