A006882 -id:A006882 - cp's OEIS frontend

A307102 Numbers written in base of double factorial numbers (A006882).

1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, 1100, 1101, 1110, 1200, 10000, 10001, 10010, 10100, 10101, 10110, 10200, 10201, 11000, 11001, 11010, 11100, 11101, 11110, 11200, 20000, 20001, 20010, 20100, 20101, 20110, 20200, 20201, 21000

Offset: 1

Sean A. Irvine, Mar 24 2019

a(1122755752855713895623244049306709034778906250) is the first term which cannot be included in the OEIS since it includes a non-decimal digit. - Charles R Greathouse IV, Sep 19 2012

Numbers in this mixed-radix number system can have multiple representations, so to avoid ambiguity this sequence assumes a greedy approach where leading digits are made as high as possible; thus we choose a(30) = 20000 rather than a(30) = 11201. - Sean A. Irvine, Mar 24 2019

			The digits (from right to left) have values 1, 2, 3, 8, 15, etc. (A006882), hence a(29) = 11200 because 29 = 1*15 + 1*8 + 2*3 + 0*2 + 0*1.

Sean A. Irvine, Table of n, a(n) for n = 1..10000
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
Sean A. Irvine, Java program (github).
F. Smarandache, Collected Papers, Vol. II.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Index entries for sequences related to factorial numbers.

Cf. A006882 (double factorial numbers), A007623 (factorial base), A019513 (erroneous version).

a[n_] := FromDigits[NumberDecompose[n, Range[n, 1, -1]!!]]; Array[a, 40] (* Amiram Eldar, May 11 2024 *)

A095159 Numerator of b(n) given by b(1) = 1, b(2) = 2; for n >= 3, b(n) = (-1)^n (2n-1) ((n-2)!!)^2/((n-1)!!)^2, where n!! is the double factorial A006882.

Original entry on oeis.org

1, 2, -5, 28, -81, 704, -325, 768, -20825, 311296, -83349, 1507328, -1334025, 3145728, -5337189, 130023424, -1366504425, 7516192768, -5466528925, 12884901888, -87470372561, 2954937499648, -349899121845, 12919261626368, -22394407746529, 52776558133248, -89580335298125

Offset: 1

N. J. A. Sloane, based on a suggestion of Leroy Quet, Jul 03 2004

b(n) is such that the continued fraction [b(1); b(2), b(3),..., b(n)] is equal to sum{k=1 to n} 1/k = H(n) = the n-th harmonic number, for all positive integers n.

a(2n)/A095175(2n) -> pi as n -> inf.; a(2n+1)/A095175(2n+1) -> -4/pi as n -> inf. - Leroy Quet, Aug 03 2004

			1, 2, -5/4, 28/9, -81/64, 704/225, -325/256, 768/245, -20825/16384, 311296/99225, ...

Cf. A006882, A095175.

A113296 Cumulative product of double factorial A006882.

Original entry on oeis.org

1, 1, 2, 6, 48, 720, 34560, 3628800, 1393459200, 1316818944000, 5056584744960000, 52563198423859200000, 2422112183371431936000000, 327312129899898454671360000000, 211155601241022491077587763200000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!! * 2!! * 3!! * 4!! * 5!! * 6!! * 7!! * 8!! * 9!! * 10!!
= 1 * 2 * 3 * 8 * 15 * 48 * 105 * 384 * 945 * 3840
= 5056584744960000 = 2^23 x 3^9 x 5^4 x 7^2.

Harvey P. Dale, Table of n, a(n) for n = 0..57
Alejandro H. Morales, Igor Pak and Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, Algebraic Combinatorics, Vol. 2, No. 5 (2019), pp. 815-861; arXiv preprint, arXiv:1707.00931 [math.CO], 2017-2020.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A006882, A074962, A137988, A137989.

Table[Product[k!!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Table[2^((6n^2+12n+2-3(-1)^n)/24) Pi^(((-1)^n-2n-3)/8) Exp[-1/8] Glaisher^(3/2) BarnesG[(2n+7+(-1)^n)/4] BarnesG[(2n+7-(-1)^n)/4], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 11 2015 *)
FoldList[Times,Range[0,20]!!] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = Product_{k=0..n} k!!.
a(n) = n!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = n*(n-2)!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = 2^((6*n^2+12*n+2-3*(-1)^n)/24) * Pi^(((-1)^n-2*n-3)/8) * exp(-1/8) * A^(3/2) * G((2n+7+(-1)^n)/4) * G((2n+7-(-1)^n)/4), where A is the Glaisher-Kinkelin constant (A074962), G(x) is the Barnes G-function. - Vladimir Reshetnikov, Nov 11 2015
Sum_{n>=0} 1/a(n) = 1/A137989. - Amiram Eldar, Nov 09 2020
Sum_{n>=0} (-1)^n/a(n) = A137988. - Amiram Eldar, Apr 12 2021

A116896 Numbers k such that k! + k!! + 1 is prime, where k!! denotes double factorial (A006882).

Original entry on oeis.org

0, 1, 2, 6, 42, 604, 1976, 5126, 14124

Offset: 1

Giovanni Resta, Mar 04 2006

1976! + 1976!! + 1 is a 5657-digit prime. Next term is greater than 4000.

			6! + 6!! + 1 = 769, which is prime.

Joe McLean, Interesting Sources of Probable Primes

Cf. A000142, A006882, A116897.

a(8) from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

a(9) from Michael S. Branicky, Nov 15 2024

A116897 Numbers k such that k! - k!! + 1 is prime, where k!! denotes double factorial (A006882).

Original entry on oeis.org

4, 6, 8, 14, 18, 28, 64, 324, 778, 882

Offset: 1

Giovanni Resta, Mar 04 2006

Next term is greater than 3000.

Next term is greater than 15000. - Michael S. Branicky, Nov 15 2024

			6! - 6!! + 1 = 673, which is prime.

Cf. A000142, A006882, A116896.

Select[Range[900],PrimeQ[#!-#!!+1]&] (* Harvey P. Dale, Mar 27 2015 *)

A129335 a(n) = phi(n!!) where phi is the Euler totient function. In other words, a(n) = A000010(A006882(n)).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 48, 128, 432, 1024, 4320, 12288, 51840, 147456, 777600, 2359296, 12441600, 42467328, 223948800, 849346560, 4702924800, 16986931200, 103464345600, 407686348800, 2586608640000, 9784472371200, 69838433280000

Offset: 1

Leroy Quet, May 26 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..808

Cf. A000010, A006882, A048855.

Table[EulerPhi[n!! ], {n, 1, 30}] (* Stefan Steinerberger, May 30 2007 *)

If n>2 is prime, a(n) = (n-1)*a(n-2). If n=2*p, where p is odd prime, a(n)=(n-2)*a(n-2). Otherwise, a(n) = n*a(n-2). - Max Alekseyev, May 26 2007

More terms from Stefan Steinerberger, May 30 2007

A129843 a(n) = number of positive integers that are <= n and are coprime to n!! (n!! = A006882(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 4, 5, 5, 5, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 7, 6, 6, 6, 7, 6, 7, 6, 7, 6, 8, 6, 8, 6, 7, 6, 8, 6, 8, 6, 8, 7, 8, 7, 9, 7, 9, 7, 10, 7, 10, 7, 10, 7, 10, 7, 11

Offset: 1

Leroy Quet, Jun 03 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006882, A056171, A070939.

f:= proc(n) local t;
     if n::odd then ilog2(n)+1
     else 1+numtheory:-pi(n) - numtheory:-pi(n/2)
     fi
end proc:
f(2):= 1:
map(f, [$1..100]); # Robert Israel, Dec 08 2022

a[n_]:=Module[{},co=0;For[i=1,iStefan Steinerberger, Jun 05 2007 *)
Table[Total[Boole[CoprimeQ[n!!,Range[n]]]],{n,80}] (* Harvey P. Dale, Dec 12 2022 *)

From Robert Israel, Dec 08 2022: (Start)
If n is odd, a(n) = A070939(n).
If n > 2 is even, a(n) = 1 + A056171(n). (End)

More terms from Stefan Steinerberger, Jun 05 2007

A153512 Triangle T(n,m) = (A006882(2n + 1))^2 / ( A006882(2m+1) * A006882(2n-2m+1) ).

Original entry on oeis.org

1, 3, 3, 15, 25, 15, 105, 245, 245, 105, 945, 2835, 3969, 2835, 945, 10395, 38115, 68607, 68607, 38115, 10395, 135135, 585585, 1288287, 1656369, 1288287, 585585, 135135, 2027025, 10135125, 26351325, 41409225, 41409225, 26351325, 10135125, 2027025

Offset: 0

Roger L. Bagula, Dec 28 2008

Row sums are 1, 6, 55, 700, 11529, 234234, 5674383, 159845400, 5136642225, 185498257230, 7438043704455...

			1;
3, 3;
15, 25, 15;
105, 245, 245, 105;
945, 2835, 3969, 2835, 945;
10395, 38115, 68607, 68607, 38115, 10395;
135135, 585585, 1288287, 1656369, 1288287, 585585, 135135;
2027025, 10135125, 26351325, 41409225, 41409225, 26351325, 10135125, 2027025;
34459425, 195270075, 585810225, 1087933275, 1329696225, 1087933275, 585810225, 195270075, 34459425;
654729075, 4146617475, 14098499415, 30211070175, 43638212475, 43638212475, 30211070175, 14098499415, 4146617475, 654729075;
13749310575, 96245174025, 365731661295, 888205463145, 1480342438575, 1749495609225, 1480342438575, 888205463145, 365731661295, 96245174025, 13749310575;

G. C. Greubel, Table of n, a(n) for n = 0..1034

T[n_, m_] = (2*n + 1)!!* Pi*Gamma[2*n + 2]/(n!*4^(n + 1)*Gamma[m + 3/ 2]*Gamma[n + 3/2 - m]);
Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

T(n,m)= A006882(2*n + 1)*Pi*Gamma(2*n + 2)/(n!*4^(n + 1)*Gamma(m + 3/2)*Gamma(n + 3/2 - m) ).

Definition replaced by an integer expression by the Assoc. Editors of the OEIS, Feb 24 2010

A172089 Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 8, 6, 8, 3, 8, 15, 20, 20, 15, 8, 15, 48, 45, 80, 45, 48, 15, 48, 105, 168, 210, 210, 168, 105, 48, 105, 384, 420, 896, 630, 896, 420, 384, 105, 384, 945, 1728, 2520, 3024, 3024, 2520, 1728, 945, 384, 945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945

Offset: 0

Roger L. Bagula, Jan 25 2010

Row sums are {1, 2, 4, 10, 28, 86, 296, 1062, 4240, 17202, 77088, ...}.

			Triangle begins
    1;
    1,    1;
    1,    2,    1;
    2,    3,    3,     2;
    3,    8,    6,     8,    3;
    8,   15,   20,    20,   15,     8;
   15,   48,   45,    80,   45,    48,   15;
   48,  105,  168,   210,  210,   168,  105,    48;
  105,  384,  420,   896,  630,   896,  420,   384,  105;
  384,  945, 1728,  2520, 3024,  3024, 2520,  1728,  945, 384;
  945, 3840, 4725, 11520, 9450, 16128, 9450, 11520, 4725, 3840, 945;

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

Cf. A000142, A006882.

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

A172089 := proc(n,m)
        factorial(n)/doublefactorial(m)/doublefactorial(n-m) ;
end proc:
seq(seq(A172089(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Oct 11 2011

binomialn[n_, k_] = n!/(Factorial2[n-k]*Factorial2[k]); Table[binomialn[n, k], {n,0,10}, {k,0,n}]//Flatten

f2(n) = prod(i=0, (n-1)\2, n - 2*i );
T(n,k) = n!/(f2(k)*f2(n-k));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 05 2019

def T(n, k): return factorial(n)/((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) = A000142(n)/(A006882(m)*A006882(n-m)).

A232701 a(n) = (2*n-1)!! mod n!, where double factorial is A006882.

Original entry on oeis.org

0, 1, 3, 9, 105, 315, 4095, 11025, 348705, 1545075, 17931375, 93087225, 3764185425, 45589819275, 1060569885375, 15877899662625, 900941666625, 5722531807867875, 90088576482279375, 1688777976676415625, 18148954872023600625, 320586579951629866875, 11054393914490520969375

Offset: 1

Alex Ratushnyak, Nov 28 2013

nonn

(2n-1)!! is the product of first n odd numbers.

			a(4) = 1*3*5*7 mod (1*2*3*4) = 105 mod 24 = 9.

Cf. A006882, A232618, A024502 (floor((2*n-1)!! / n!)).

o = 1; Reap[For[n = 1, n <= 99, n += 2, o *= n; m = Mod[o, (Quotient[n, 2] + 1)!]; Sow[m]]][[2, 1]] (* Jean-François Alcover, Oct 05 2017, translated from Alex Ratushnyak's Python code *)

import math
o=1
for n in range(1,99,2):
  o*=n
  print(o % math.factorial(n//2+1), end=', ')

cp's OEIS Frontend

A307102 Numbers written in base of double factorial numbers (A006882).

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A095159 Numerator of b(n) given by b(1) = 1, b(2) = 2; for n >= 3, b(n) = (-1)^n (2n-1) ((n-2)!!)^2/((n-1)!!)^2, where n!! is the double factorial A006882.

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A113296 Cumulative product of double factorial A006882.

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A116896 Numbers k such that k! + k!! + 1 is prime, where k!! denotes double factorial (A006882).

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A116897 Numbers k such that k! - k!! + 1 is prime, where k!! denotes double factorial (A006882).

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Mathematica

A129335 a(n) = phi(n!!) where phi is the Euler totient function. In other words, a(n) = A000010(A006882(n)).

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A129843 a(n) = number of positive integers that are <= n and are coprime to n!! (n!! = A006882(n)).

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A153512 Triangle T(n,m) = (A006882(2*n + 1))^2 / ( A006882(2*m+1) * A006882(2*n-2*m+1) ).

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A172089 Triangle T(n,m) = n!/(m!!*(n-m)!!) read by rows, where (.)!! = A006882(.) are double factorials.

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A153512 Triangle T(n,m) = (A006882(2n + 1))^2 / ( A006882(2m+1) * A006882(2n-2m+1) ).