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A319545 a(n) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14*15 - ... + (up to n). 9

%I #17 Mar 30 2023 17:07:37

%S 1,2,6,24,120,114,78,-216,-2904,-30120,-30109,-29988,-28404,-6096,

%T 330240,330224,329968,325344,237216,-1530240,-1530219,-1529778,

%U -1519614,-1275216,4845360,4845334,4844658,4825704,4275336,-12255360,-12255329,-12254368,-12222624

%N a(n) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14*15 - ... + (up to n).

%C 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=5.

%C An alternating version of A319206.

%H Harvey P. Dale, <a href="/A319545/b319545.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (-1)^floor(n/5) * Sum_{i=1..4} (1-sign((n-i) mod 5)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/5)+1) * (1-sign(i mod 5)) * (Product_{j=1..5} (i-j+1)).

%e a(1) = 1;

%e a(2) = 1*2 = 2;

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

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

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

%e a(6) = 1*2*3*4*5 - 6 = 114;

%e a(7) = 1*2*3*4*5 - 6*7 = 78;

%e a(8) = 1*2*3*4*5 - 6*7*8 = -216;

%e a(9) = 1*2*3*4*5 - 6*7*8*9 = -2904;

%e a(10) = 1*2*3*4*5 - 6*7*8*9*10 = -30120;

%e a(11) = 1*2*3*4*5 - 6*7*8*9*10 + 11 = -30109;

%e a(12) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12 = -29988;

%e a(13) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13 = -28404;

%e a(14) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14 = -6096;

%e a(15) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14*15 = 330240;

%e a(16) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14*15 - 16 = 330224;

%e a(17) = 1*2*3*4*5 - 6*7*8*9*10 + 11*12*13*14*15 - 16*17 = 329968; etc.

%t a[n_]:=(-1)^Floor[n/5]*Sum[(1-Sign[Mod[n-i,5]])*Product[n-j+1,{j,1,i}],{i,1,4}]+Sum[(-1)^(Floor[i/5]+1)*(1-Sign[Mod[i,5]])*Product[i-j+1,{j,1,4}],{i,1,n}]; Array[a, 30] (* _Stefano Spezia_, Sep 23 2018 *)

%t Table[Total[Times@@@Partition[Riffle[Times@@@Partition[Range[n],UpTo[5]],{1,-1},{2,-1,2}],2]],{n,40}] (* _Harvey P. Dale_, Mar 30 2023 *)

%Y For similar sequences, see: A001057 (k=1), A319373 (k=2), A319543 (k=3), A319544 (k=4), this sequence (k=5), A319546 (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).

%Y Cf. A319206.

%K sign,easy

%O 1,2

%A _Wesley Ivan Hurt_, Sep 22 2018

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Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)