A275062 -id:A275062 - cp's OEIS frontend

A010551 Multiply successively by 1,1,2,2,3,3,4,4,..., n >= 1, a(0) = 1.

1, 1, 1, 2, 4, 12, 36, 144, 576, 2880, 14400, 86400, 518400, 3628800, 25401600, 203212800, 1625702400, 14631321600, 131681894400, 1316818944000, 13168189440000, 144850083840000, 1593350922240000, 19120211066880000, 229442532802560000, 2982752926433280000

Offset: 0

Mark R. Diamond

From Emeric Deutsch, Dec 14 2008: (Start)
Number of permutations of {1,2,...,n-1} having a single run of odd entries. Example: a(5)=12 because we have 1324,1342,3124,3142,2134,4132,2314,4312, 2413, 4213, 2431 and 4231.
a(n) = A152666(n-1,1). (End)
a(n+1) gives the permanent of the n X n matrix whose (i,j)-element is i+j-1 modulo 2. - John W. Layman, Jan 03 2011
From Daniel Forgues, May 20 2011: (Start)
a(0) = 1 since it is the empty product.
A010551(n-2), n >= 2, equal to (ceiling((n-2)/2))! * (floor((n-2)/2))!, gives the number of arrangements of n-2 entries from 2 to n-1, starting with an even entry and where the parity of adjacent entries alternates. This is the number of arrangements to investigate for row n of a prime pyramid (A051237). (End)
Partial products of A008619. - Reinhard Zumkeller, Apr 02 2012
Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb where a < b < c, cf. A210667 (equivalently under such transformations of the form abc <--> bac where a < b < c.) - Tom Roby, May 15 2012
Row sums of A246117. - Peter Bala, Aug 15 2014
a(n) is the number of parity-alternating permutations of size n. A permutation is parity-alternating if it sends even integers to even, and odd to odd. - Per W. Alexandersson, Jun 06 2022
n divides a(n) if and only if n is not prime. Since a(n) = floor(n/2)!*floor((n+1)/2)!, if n is prime then n is not a factor of a(n). All the prime factors of a(n) are in fact less than or equal to (n+1)/2. If n is composite, then it's possible to write it as p*q with p and q less than or equal to n/2. So p and q are factors of a(n). - Davide Oliveri, Apr 01 2023
Number of permutations of {1, 2, ..., n-1} where each entry is not greater than twice the previous entry. - Dewangga Putra Sheradhien, Jul 13 2024

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 144*x^7 + 576*x^8 + ...
For n = 7, a(n) = 1*1*2*2*3*3*4 (7 factors), which is 144. - _Michael B. Porter_, Jul 03 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Edinah K. Gnang and Isaac Wass, Growing Graceful and Harmonious Trees, arXiv:1808.05551 [math.CO], 2018-2020. See proposition 1.
Frether Getachew Kebede and Fanja Rakotondrajao, Parity alternating permutations starting with an odd integer, arXiv:2101.09125 [math.CO], 2021.
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.

Cf. A008619, A064044, A246117, A130820, A229020.

Column k=2 of A275062.

a010551 n = a010551_list !! n
a010551_list = scanl (*) 1 a008619_list
-- Reinhard Zumkeller, Apr 02 2012

[Factorial(n div 2)*Factorial((n+1) div 2): n in [0..25]]; // Vincenzo Librandi Jan 17 2018

A010551 := proc(n)
    option remember;
    if n <= 1 then
        1
    else
        procname(n-1) *trunc( (n+1)/2 );
    fi;
end:

FoldList[ Times, 1, Flatten@ Array[ {#, #} &, 11]] (* Robert G. Wilson v, Jul 14 2010 *)

{a(n)=local(X=x+x*O(x^n)); 1/polcoeff(besseli(0,2*X)+X*besseli(1,2*X),n,x)} \\ Paul D. Hanna, Apr 07 2005

A010551(n)=(n\2)!*((n+1)\2)! \\ Michael Somos, Dec 29 2012, edited by M. F. Hasler, Nov 26 2017

def O(f):
    c = 1
    while len(f) > 1:
        f.sort()
        m = abs(f[0] - f[1])
        c *= m
        f[0] = m
        f.pop(1)
    return c
a = lambda n: O(list(range(1, n+1)))
print([a(n) for n in range(0, 26)]) # Darío Clavijo, Aug 24 2024

a(n) = floor(n/2)!*floor((n+1)/2)! is the number of permutations p of {1, 2, 3, ..., n} such that for every i, i and p(i) have the same parity, i.e., p(i) - i is even. - Avi Peretz (njk(AT)netvision.net.il), Feb 22 2001
a(n) = n!/binomial(n, floor(n/2)). - Paul Barry, Sep 12 2004
G.f.: Sum_{n>=0} x^n/a(n) = besseli(0, 2*x) + x*besseli(1, 2*x). - Paul D. Hanna, Apr 07 2005
E.g.f.: 1/(1-x/2) + (1/2)/(1-x/2)*arccos(1-x^2/2)/sqrt(1-x^2/4). - Paul D. Hanna, Aug 28 2005
G.f.: G(0) where G(k) = 1 + (k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 28 2012
D-finite with recurrence: 4*a(n) - 2*a(n-1) - n*(n-1)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012
a(n) = a(n-1) * (a(n-2) + a(n-3)) / a(n-3) for all n >= 3. - Michael Somos, Dec 29 2012
G.f.: 1 + x + x^2*(1 + x*(G(0) - 1)/(x-1)) where G(k) = 1 - (k+2)/(1-x/(x - 1/(1 - (k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-x/(x - 1/(1 - (k+1)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*G(0), where G(k) = 1 + x*(k+1)/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
G.f.: Q(0), where Q(k) = 1 + x*(k+1)/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013
Sum_{n >= 1} 1/a(n) = A130820. - Peter Bala, Jul 02 2016
a(n) ~  sqrt(Pi*n) * n! / 2^(n + 1/2). - Vaclav Kotesovec, Oct 02 2018
Sum_{n>=0} (-1)^n/a(n) = A229020. - Amiram Eldar, Apr 12 2021

A264557 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 24, 72, 216, 864, 3456, 13824, 69120, 345600, 1728000, 10368000, 62208000, 373248000, 2612736000, 18289152000, 128024064000, 1024192512000, 8193540096000, 65548320768000, 589934886912000, 5309413982208000, 47784725839872000

Offset: 1

R. H. Hardin, Nov 17 2015

nonn

Column 1 of A264560.

			All solutions for n=4
..0....3
..3....0
..1....1
..2....2

R. H. Hardin, Table of n, a(n) for n = 1..48

Cf. A264560.

Column k=3 of A275062.

Table[Product[Floor[(n + i)/3]!, {i, 0, 2}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..2} ceiling((n-i)/3)!. - Alois P. Heinz, Jul 09 2016

a(n) ~ 2 * Pi * (n+1)! / 3^(n + 3/2). - Vaclav Kotesovec, Oct 02 2018

A264635 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 48, 144, 432, 1296, 5184, 20736, 82944, 331776, 1658880, 8294400, 41472000, 207360000, 1244160000, 7464960000, 44789760000, 268738560000, 1881169920000, 13168189440000, 92177326080000, 645241282560000, 5161930260480000

Offset: 1

R. H. Hardin, Nov 19 2015

nonn

			All solutions for n=6:
  0    4    4    0
  4    0    0    4
  1    1    5    5
  5    5    1    1
  2    2    2    2
  3    3    3    3

R. H. Hardin, Table of n, a(n) for n = 1..62

Column 1 of A264638.

Column k=4 of A275062.

Table[Product[Floor[(n + i)/4]!, {i, 0, 3}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..3} floor((n+i)/4)!. - Alois P. Heinz, Jul 12 2016
a(n) ~ Pi^(3/2) * n^(3/2) * n! / 2^(2*n + 5/2). - Vaclav Kotesovec, Oct 02 2018
Sum_{n>0} floor((n-1)/4)/a(n) = 1. - Peter McNair, May 29 2022

A264656 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 96, 288, 864, 2592, 7776, 31104, 124416, 497664, 1990656, 7962624, 39813120, 199065600, 995328000, 4976640000, 24883200000, 149299200000, 895795200000, 5374771200000, 32248627200000, 193491763200000, 1354442342400000

Offset: 1

R. H. Hardin, Nov 20 2015

nonn

Column 1 of A264659.

			All solutions for n=8
..5....0....0....5....5....5....0....0
..0....5....5....0....0....0....5....5
..1....6....6....6....6....1....1....1
..6....1....1....1....1....6....6....6
..7....7....2....2....7....2....7....2
..2....2....7....7....2....7....2....7
..3....3....3....3....3....3....3....3
..4....4....4....4....4....4....4....4

R. H. Hardin, Table of n, a(n) for n = 1..74

Cf. A264659.

Column k=5 of A275062.

Table[Product[Floor[(n + i)/5]!, {i, 0, 4}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..4} floor((n+i)/5)!. - Alois P. Heinz, Jul 12 2016

a(n) ~ (2*Pi*n)^2 * n! / 5^(n + 5/2). - Vaclav Kotesovec, Oct 02 2018

A264701 Number of n X 1 arrays of permutations of 0..n-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 192, 576, 1728, 5184, 15552, 46656, 186624, 746496, 2985984, 11943936, 47775744, 191102976, 955514880, 4777574400, 23887872000, 119439360000, 597196800000, 2985984000000, 17915904000000, 107495424000000, 644972544000000

Offset: 1

R. H. Hardin, Nov 21 2015

nonn

Column 1 of A264704.

			All solutions for n=8
..6....6....0....0
..0....0....6....6
..7....1....1....7
..1....7....7....1
..2....2....2....2
..3....3....3....3
..4....4....4....4
..5....5....5....5

R. H. Hardin, Table of n, a(n) for n = 1..84

Cf. A264704.

Column k=6 of A275062.

Table[Product[Floor[(n + i)/6]!, {i, 0, 5}], {n, 1, 40}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..5} floor((n+i)/6)!. - Alois P. Heinz, Jul 12 2016

a(n) ~ (2*Pi*n)^(5/2) * n! / 6^(n + 3). - Vaclav Kotesovec, Oct 02 2018

A264791 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 1119744, 4478976, 17915904, 71663616, 286654464, 1146617856, 4586471424, 22932357120, 114661785600, 573308928000, 2866544640000, 14332723200000, 71663616000000

Offset: 1

R. H. Hardin, Nov 25 2015

nonn

Column 1 of A264794.

			All solutions for n=11
..0....0....7....0....7....0....7....7....7....7....0....0....0....7....0....7
..7....7....0....7....0....7....0....0....0....0....7....7....7....0....7....0
..1....1....8....8....8....1....1....8....1....8....8....8....8....1....1....1
..8....8....1....1....1....8....8....1....8....1....1....1....1....8....8....8
..9....2....2....9....9....2....9....2....2....9....9....2....2....9....9....2
..2....9....9....2....2....9....2....9....9....2....2....9....9....2....2....9
..3....3...10....3....3...10...10....3...10...10...10...10....3....3...10....3
.10...10....3...10...10....3....3...10....3....3....3....3...10...10....3...10
..4....4....4....4....4....4....4....4....4....4....4....4....4....4....4....4
..5....5....5....5....5....5....5....5....5....5....5....5....5....5....5....5
..6....6....6....6....6....6....6....6....6....6....6....6....6....6....6....6

R. H. Hardin, Table of n, a(n) for n = 1..91

Cf. A264794.

Column k=7 of A275062.

Table[Product[Floor[(n + i)/7]!, {i, 0, 6}], {n, 1, 40}] (* Vaclav Kotesovec, Oct 02 2018 *)

a(n) = Product_{i=0..6} floor((n+i)/7)!. - Alois P. Heinz, Jul 12 2016

a(n) ~ (2*Pi*n)^3 * n! / 7^(n + 7/2). - Vaclav Kotesovec, Oct 02 2018

A317980 a(n) = Product_{i=1..n} floor(5*i/2).

Original entry on oeis.org

2, 10, 70, 700, 8400, 126000, 2142000, 42840000, 942480000, 23562000000, 636174000000, 19085220000000, 610727040000000, 21375446400000000, 790891516800000000, 31635660672000000000, 1328697748224000000000, 59791398670080000000000, 2810195737493760000000000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

If p > 2 and p is odd, then Product_{i=1..n} floor(p*i/2) ~ (p/2)^n * n! * 2^(1/(2*p)) * sqrt(Pi) / (Gamma(1/2 - 1/(2*p)) * n^(1/(2*p))).

Harvey P. Dale, Table of n, a(n) for n = 1..390

Cf. A010786, A180736, A275062, A319948, A319950.

Table[Product[Floor[i*5/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 10 a[n - 1] - 5 (n - 1) (5 n - 6) a[n - 2] == 0, a[1] == 2, a[2] == 10}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)
FoldList[Times,Floor[5*Range[20]/2]] (* Harvey P. Dale, Sep 17 2020 *)

a(n) ~ (5/2)^n * n! * 2^(1/10) * sqrt(Pi) / (Gamma(2/5) * n^(1/10)).

Recurrence: 4*a(n) - 10*a(n-1) - 5*(n - 1)*(5*n - 6)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319948 a(n) = Product_{i=1..n} floor(3*i/2).

Original entry on oeis.org

1, 3, 12, 72, 504, 4536, 45360, 544320, 7076160, 106142400, 1698278400, 30569011200, 580811212800, 12197035468800, 268334780313600, 6440034727526400, 161000868188160000, 4347023441080320000, 121716656350248960000, 3651499690507468800000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

Cf. A010786, A180736, A275062, A319949, A319950, A317980.

Table[Product[Floor[i*3/2], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{4 a[n] - 6 a[n - 1] - 3 (n - 1) (3 n - 4) a[n - 2] == 0, a[1] == 1, a[2] == 3}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)

a(n) ~ (3/2)^n * n! * 2^(1/6) * sqrt(Pi) / (Gamma(1/3) * n^(1/6)).

Recurrence: 4*a(n) - 6*a(n-1) - 3*(n - 1)*(3*n - 4)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319949 a(n) = Product_{i=1..n} floor(4*i/3).

Original entry on oeis.org

1, 2, 8, 40, 240, 1920, 17280, 172800, 2073600, 26956800, 377395200, 6038323200, 102651494400, 1847726899200, 36954537984000, 776045297664000, 17072996548608000, 409751917166592000, 10243797929164800000, 266338746158284800000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..429

Cf. A004773, A010786, A180736, A275062, A319948, A319950.

Table[Product[Floor[i*4/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(3*n - 7)*a[n] == 54*(2*n - 5)*a[n-1] + 12*(12*n^2 - 42*n + 35)*a[n-2] + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a[n-3], a[1]==1, a[2]==2, a[3]==8}, a, {n, 1, 20}]
FoldList[Times,Floor[4 Range[20]/3]] (* Harvey P. Dale, Mar 21 2024 *)

a(n) = prod(i=1, n, (4*i)\3); \\ Michel Marcus, Oct 03 2018

a(n) ~ (4/3)^n * n! * 2*sqrt(Pi) / (3^(1/4) * Gamma(1/4) * n^(1/4)).

Recurrence: 27*(3*n - 7)*a(n) = 54*(2*n - 5)*a(n-1) + 12*(12*n^2 - 42*n + 35)*a(n-2) + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a(n-3).

A319950 a(n) = Product_{i=1..n} floor(5*i/3).

Original entry on oeis.org

1, 3, 15, 90, 720, 7200, 79200, 1029600, 15444000, 247104000, 4447872000, 88957440000, 1868106240000, 42966443520000, 1074161088000000, 27928188288000000, 781989272064000000, 23459678161920000000, 727250023019520000000, 23999250759644160000000

Offset: 1

Vaclav Kotesovec, Oct 02 2018

nonn

If p > 3 and gcd(p,3)=1 then Product_{i=1..n} floor(i*p/3) ~ (p/3)^n * n! * 2*Pi * 3^(1/p - 1/2) / (c(p) * n^(1/p)), where
c(p) = Gamma(2/3 - 2/(3*p)) * Gamma(1/3 - 1/(3*p)) if mod(p, 3) = 1,
c(p) = Gamma(1/3 - 2/(3*p)) * Gamma(2/3 - 1/(3*p)) if mod(p, 3) = 2.
In general, if q > 1, p > q and gcd(p,q)=1, then Product_{i=1..n} floor(i*p/q) ~ c(p,q) * (p/q)^n * n! / n^((q-1)/(2*p)), where c(p,q) is a constant.

Cf. A010786, A047220, A180736, A275062, A319948, A319949, A317980.

Table[Product[Floor[i*5/3], {i, 1, n}], {n, 1, 20}]
RecurrenceTable[{27*(15*n - 32)*a[n] == 675*(n-2)*a[n-1] + 15*(75*n^2 - 255*n + 194)*a[n-2] + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a[n-3], a[1]==1, a[2]==3, a[3]==15}, a, {n, 1, 20}]

a(n) = prod(i=1, n, (5*i)\3); \\ Michel Marcus, Oct 03 2018

a(n) ~ (5/3)^n * n! * 2*Pi / (3^(3/10) * Gamma(1/5) * Gamma(3/5) * n^(1/5)).

Recurrence: 27*(15*n - 32)*a(n) = 675*(n-2)*a(n-1) + 15*(75*n^2 - 255*n + 194)*a(n-2) + 5*(n-2)*(5*n - 12)*(5*n - 11)*(15*n - 17)*a(n-3).

cp's OEIS Frontend

A010551 Multiply successively by 1,1,2,2,3,3,4,4,..., n >= 1, a(0) = 1.

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A264557 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 3.

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A264635 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 4.

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A264656 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 5.

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A264701 Number of n X 1 arrays of permutations of 0..n-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 6.

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A264791 Number of n X 1 arrays of permutations of 0..n*1-1 with rows nondecreasing modulo 2 and columns nondecreasing modulo 7.

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A317980 a(n) = Product_{i=1..n} floor(5*i/2).

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