A129981 -id:A129981 - cp's OEIS frontend

A176788 a(n) = (0!! + 1!! + 2!! + 3!! + ... + (n-1)!!) mod n.

0, 0, 1, 3, 0, 0, 1, 7, 0, 2, 6, 3, 12, 10, 12, 3, 13, 6, 11, 7, 3, 8, 14, 15, 12, 22, 15, 3, 19, 12, 16, 11, 30, 14, 17, 15, 17, 34, 9, 7, 26, 24, 36, 19, 42, 8, 2, 3, 24, 12, 48, 35, 2, 42, 52, 31, 15, 22, 51, 27, 32, 56, 24, 27, 22, 30, 6, 31, 54, 52, 12, 15, 7, 56, 12, 15, 52, 48

Offset: 1

Paolo P. Lava & Giorgio Balzarotti, Apr 26 2010

			0!! mod 1 = 1 mod 1 = 0.
(0!!+1!!) mod 2 = (1 + 1) mod 2 = 0.
(0!!+1!!+2!!) mod 3 = (1 + 1 + 2) mod 3 = 1.

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

Cf. A049782, A176787, A176789.

A176788 := proc(n)
        A129981(n-1) mod n ;
end proc: # R. J. Mathar, Jul 13 2012

Table[Mod[Total[Range[0,n-1]!!],n],{n,80}] (* Harvey P. Dale, Nov 15 2021 *)

a(n) = A129981(n-1) mod n.

A172088 Triangle: T(n,m) = n!! - m!! - (n-m)!! read by rows 0 <= m <= n, where ()!! are the double factorials.

Original entry on oeis.org

-1, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, 4, 4, 4, -1, -1, 6, 10, 10, 6, -1, -1, 32, 38, 42, 38, 32, -1, -1, 56, 88, 94, 94, 88, 56, -1, -1, 278, 334, 366, 368, 366, 334, 278, -1, -1, 560, 838, 894, 922, 922, 894, 838, 560, -1, -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1

Offset: 0

Roger L. Bagula, Jan 25 2010

Row sums are {-1, -2, -2, -2, 10, 30, 180, 474, 2322, 6426, 31536, ...}; n-th row sum is (n+1)*n!! - 2*A129981(n).

			Triangle begins
  -1;
  -1,   -1;
  -1,    0,   -1;
  -1,    0,    0,   -1;
  -1,    4,    4,    4,   -1;
  -1,    6,   10,   10,    6,   -1;
  -1,   32,   38,   42,   38,   32,   -1;
  -1,   56,   88,   94,   94,   88,   56,   -1;
  -1,  278,  334,  366,  368,  366,  334,  278,   -1;
  -1,  560,  838,  894,  922,  922,  894,  838,  560,   -1;
  -1, 2894, 3454, 3732, 3784, 3810, 3784, 3732, 3454, 2894, -1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A006882, A129981.

F2:=func< n | &*[n..2 by -2] >;
[F2(n) - F2(k) - F2(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 05 2019

A172088 := proc(n,m)
        doublefactorial(n)-doublefactorial(m)-doublefactorial(n-m) ;
end proc:
seq(seq(A172088(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 11 2011

T[n_, k_] = n!! -k!! -(n-k)!!; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

f2(n) = prod(j=0, (n-1)\2, n-2*j);
T(n,k) = f2(n) - f2(k) - f2(n-k); \\ G. C. Greubel, Dec 05 2019

def T(n, k): return (n).multifactorial(2) - (k).multifactorial(2) - (n-k).multifactorial(2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 05 2019

T(n,m) = A006882(n) - A006882(m) - A006882(n-m).

cp's OEIS Frontend

A176788 a(n) = (0!! + 1!! + 2!! + 3!! + ... + (n-1)!!) mod n.

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