A319550 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 - ... + (up to n).

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 362870, 362770, 361560, 345720, 122640, -3240720, -57294720, -979816320, -17642862720, -17642862701, -17642862340, -17642854740, -17642687160, -17638824840, -17545953600, -15220134720, 45348065280, 1683112193280

Offset: 1

Author

Wesley Ivan Hurt, Sep 22 2018

Keywords

• sign
• easy

Comments

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=9.

An alternating version of A319211.

Examples

			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 = 362870;
a(11) = 1*2*3*4*5*6*7*8*9 - 10*11 = 362770;
a(12) = 1*2*3*4*5*6*7*8*9 - 10*11*12 = 361560;
a(13) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13 = 345720;
a(14) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14 = 122640;
a(15) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15 = -3240720;
a(16) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16 = -57294720;
a(17) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17 = -979816320;
a(18) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17*18 = -17642862720;
a(19) = 1*2*3*4*5*6*7*8*9 - 10*11*12*13*14*15*16*17*18 + 19 = -17642862701; etc.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

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), this sequence (k=9), A319551 (k=10).

Cf. A319211.

Programs

Mathematica

Table[Total[Times@@@Partition[Riffle[Times@@@Partition[Range[n],UpTo[9]],{1,-1},{1,-1,2}],2]],{n,30}] (* Harvey P. Dale, Oct 05 2024 *)

Formula

a(n) = (-1)^floor(n/9) * Sum_{i=1..8} (1-sign((n-i) mod 9)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/9)+1) * (1-sign(i mod 9)) * (Product_{j=1..9} (i-j+1)).