A007911 -id:A007911 - cp's OEIS frontend

A007912 Quantum factorials: (n-1)!! - (n-2)!! (mod n).

1, 1, 0, 1, 5, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 9, 2, 15, 0, 1, 2, 9, 0, 1, 0, 7, 0, 15, 2, 1, 0, 1, 0, 27, 0, 1, 0, 25, 0, 21, 2, 11, 0, 1, 45, 33, 0, 25, 0, 39, 0, 27, 0, 49, 0, 1, 57, 15, 0, 1, 0, 1, 0, 33, 2, 51, 0, 35, 2, 9, 0, 1, 0, 19, 0, 39, 77, 65, 0, 1, 81, 63, 0, 1, 0, 33, 0, 45, 0

Offset: 3

J. H. Conway

S. P. Hurd and J. S. McCranie, Quantum factorials. Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994). Congr. Numer. 104 (1994), 19-24.

T. D. Noe, Table of n, a(n) for n = 3..2000
S. P. Hurd and J. S. McCranie, Quantum factorials, 1994.

Cf. A006882, A007911.

a:= n-> (d-> irem(d(n-1)-d(n-2), n))(doublefactorial):
seq(a(n), n=3..100);  # Alois P. Heinz, Dec 17 2021

Table[Mod[(n-1)!!-(n-2)!!,n],{n,3,100}] (* Harvey P. Dale, Aug 07 2012 *)

a(n) = 0 iff n is odd and not a prime congruent to 3 modulo 4. - Charlie Neder, Feb 24 2019

A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 8, 7, 9, 1, 1, 15, 33, 12, 14, 1, 1, 48, 57, 87, 18, 20, 1, 1, 105, 279, 141, 185, 25, 27, 1, 1, 384, 561, 975, 285, 345, 33, 35, 1, 1, 945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1, 3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1

Offset: 0

Philippe Deléham, Oct 28 2013

Triangle A180048 mixed with triangle A180049.

Let p(n,x) be the polynomial whose coefficients are given by row n; e.g., p(2,x) = 2 + x + x^2; then p(n,x) is the numerator of the rational function given by f(n,x) = x + (n - 1)/f(n-1,x), where f(x,0) = 1. (Sum of numbers in row n) = A000885(n) for n >= 1. (Column 1) = A006882 (n-th term = n!! for n >= 0) - Clark Kimberling, Oct 19 2014

			Triangle begins (0<=k<=n):
1
1, 1
2, 1, 1
3, 5, 1, 1
8, 7, 9, 1, 1
15, 33, 12, 14, 1, 1
48, 57, 87, 18, 20, 1, 1
105, 279, 141, 185, 25, 27, 1, 1
384, 561, 975, 285, 345, 33, 35, 1, 1
945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1
3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
10395, 35685, 26685, 41685, 10290, 12558, 1302, 1422, 63, 65, 1, 1

Clark Kimberling, Rows 0..100, flattened

Cf. A180048, A180049.

t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k > n || k < 0, 0, t[n - 1, k - 1] + n*t[n - 2, k]]; Table[t[n, k], {n, 0, 10}, {k, 0, n}](* Clark Kimberling, Oct 19 2014 *)
(* Next, the polynomials *); z = 20; f[x_, n_] := x + n/f[x, n - 1]; f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}]; u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]]  (* A249057 array *)
Flatten[CoefficientList[u, x]] (* A249057 sequence *)
(* Clark Kimberling, Oct 19 2014 *)

T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.
T(n,0) = A006882(n).
T(n+1,1) = A007911(n+3).
Sum_{k=0..n} T(n,k) = A000085(n+1).

Corrected by Clark Kimberling, Oct 21 2014

A204912 Ordered differences of double factorials.

Original entry on oeis.org

1, 2, 1, 7, 6, 5, 14, 13, 12, 7, 47, 46, 45, 40, 33, 104, 103, 102, 97, 90, 57, 383, 382, 381, 376, 369, 336, 279, 944, 943, 942, 937, 930, 897, 840, 561, 3839, 3838, 3837, 3832, 3825, 3792, 3735, 3456, 2895, 10394, 10393, 10392, 10387, 10380

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

As a triangle with rows {1}, {2,1}, {7,6,5}, ..., evidently the first diagonal is A007911 = {1,1,5,7,33,57,279,...} (up to an offset). - L. Edson Jeffery, Jan 24 2012

			a(1)=2!!-1!!=2-1=1
a(2)=3!!-1!!=3-1=2
a(3)=3!!-2!!=3-2=1
a(4)=4!!-1!!=8-1=7
a(5)=4!!-2!!=8-2=6

Cf. A204982, A204892.

Cf. A007911.

Mathematica
```
(See the program at A204982.)
```

A227415 a(n) = (n+1)!! mod n!!.

Original entry on oeis.org

0, 0, 1, 2, 7, 3, 9, 69, 177, 60, 2715, 4500, 42975, 104580, 91665, 186795, 3493665, 13497435, 97345395, 442245825, 2601636975, 13003053525, 70985324025, 64585694250, 57891366225, 3576632909850, 9411029102475, 147580842959550, 476966861546175, 5708173568847750

Offset: 0

Alex Ratushnyak, Jul 10 2013

a(n) is divisible by A095987(n+1), and is nonzero for n > 1. - Robert Israel, Mar 10 2016

			a(4) = 5*3 mod 4*2 = 15 mod 8 = 7.

Robert Israel, Table of n, a(n) for n = 0..800

Cf. A006882, A095987.
Cf. A007911: (n-1)!! - (n-2)!!
Cf. A007912: (n-1)!! - (n-2)!! (mod n).
Cf. A060696: (n-1)!! + (n-2)!! except first two terms.

seq(doublefactorial(n+1) mod doublefactorial(n), n=0..100); # Robert Israel, Mar 10 2016

for n in range(2, 77):
    prOdd = prEven = 1
    for i in range(1, n, 2): prOdd *= i
    for i in range(2, n, 2): prEven *= i
    if n&1: print(prEven % prOdd, end=', ')
    else:   print(prOdd % prEven, end=', ')

a(n) = A006882(n+1) mod A006882(n).

cp's OEIS Frontend

A007912 Quantum factorials: (n-1)!! - (n-2)!! (mod n).

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A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

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A204912 Ordered differences of double factorials.

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A227415 a(n) = (n+1)!! mod n!!.

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