A003149 -id:A003149 - cp's OEIS frontend

A046826 Denominator of Sum_{k=0..n} 1/binomial(n,k).

1, 1, 2, 3, 3, 5, 60, 105, 35, 63, 630, 1155, 6930, 12870, 24024, 9009, 9009, 17017, 306306, 2909907, 692835, 1322685, 58198140, 111546435, 66927861, 128707425, 371821450, 717084225, 20078358300, 38818159380, 2329089562800, 4512611027925

Offset: 0

N. J. A. Sloane

			1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826

See A046825, which is the main entry.

T. D. Noe, Table of n, a(n) for n=0..200

Cf. A003149, A046825.

[Denominator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021

Denominator[Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}]] (* Harvey P. Dale, Nov 05 2011 *)

[denominator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021

a(n) = denominator( A003149(n)/n! ). - G. C. Greubel, May 24 2021

A126674 a(n) = n!*Sum_{j=0..n-1} 2^j/(j+1).

Original entry on oeis.org

0, 1, 4, 20, 128, 1024, 9984, 115968, 1572864, 24477696, 430571520, 8452177920, 183175741440, 4343275192320, 111817607086080, 3105593229312000, 92539365359616000, 2944365169213440000, 99619235621240832000, 3571109329517936640000, 135199252993504444416000, 5390266968989421797376000

Offset: 0

N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

nonn

R. J. Mathar's recurrence is correct. a(n) has a new sum term in addition to what a(n-1) has, giving a(n) = n*a(n-1) + 2^(n-1)*(n-1)!. (Cf. A000165 = 2^n*n!.) The same for a(n-1) from a(n-2), and a factor, is 2*(n-1)*(a(n-1) - (n-1)*a(n-2)) = 2^(n-1)*(n-1)! too. Substitute it leaving a(n) in terms of a(n-1) and a(n-2). The recurrence shows the o.g.f. satisfies the differential equation (2*x^2-x+1)*g + 3*x^2*(2*x-1)*g' + 2*x^4*g'' - x = 0. - Kevin Ryde, Jul 11 2019

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Row sums of A126671.

[0] cat [Factorial(n)*(&+[2^j/(j+1):j in [0..n-1]]):n in [1..21]]; // Marius A. Burtea, Jul 12 2019

Maple
```
F:=n->add( n!*2^i/(1+i), i=0..n-1);
```

Table[n!Sum[2^j/(j+1),{j,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jun 14 2017 *)

a(n) = n!*sum(j=0, n-1, 2^j/(j+1)); \\ Michel Marcus, Jul 12 2019

From Vladeta Jovovic, Feb 13 2007: (Start)
a(n) = 2^(n-1)*A003149(n-1).
O.g.f.: x*(Sum_{k>=0} k!*(2*x)^k)^2.
E.g.f.: log(1-2*x)/(x-1)/2. (End)
E.g.f.: E(x) = 1/2*log(1 - 2*x)/(x - 1) = x*(1 - x*G(0))/(x-1)/(2*x-1); G(k) = 1 + 2*x*(2*k+1)/(2*k + 3 - 2*x*(k+1)*(2*k+3)/(2*x*(k+1) + (k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 13 2011
G.f.: x*hypergeom([1,1],[],2*x)^2. - Mark van Hoeij, May 16 2013
Conjecture: a(n) + (-3*n+2)*a(n-1) + 2*(n-1)^2*a(n-2) = 0. - R. J. Mathar, May 23 2014
G.f.: x*(1/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...))))))))^2. - Ilya Gutkovskiy, May 10 2017

A136128 Number of components in all permutations of [1,2,...,n].

Original entry on oeis.org

1, 3, 10, 40, 192, 1092, 7248, 55296, 478080, 4625280, 49524480, 581368320, 7422589440, 102372076800, 1516402944000, 24004657152000, 404347023360000, 7220327288832000, 136227009945600000, 2707657158721536000, 56546150835879936000, 1237826569587277824000

Offset: 1

Emeric Deutsch, Jan 21 2008

nonn

			a(3) = 10 because the permutations of [1,2,3], with components separated by /, are 1/2/3, 1/32, 21/3, 231, 312 and 321.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

Alois P. Heinz, Table of n, a(n) for n = 1..450
Yujia Kang, Thomas Selig, Guanyi Yang, Yanting Zhang, and Haoyue Zhu, On friendship and cyclic parking functions, arXiv:2310.06560 [math.CO], 2023-2024. See p. 13.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.

Cf. A003149, A059371, A059438.

seq(add(factorial(i)*factorial(n-i),i=0..n-1),n=1..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      (a(n-1)+(n-1)!)*(n+1)/2)
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 13 2019

nn=20; p=Sum[n!x^n,{n,0,nn}]; Drop[CoefficientList[Series[p(p-1), {x,0,nn}], x], 1] (* Geoffrey Critzer, Apr 20 2012 *)
Table[(n + 1)! Re[-LerchPhi[2, 1, n + 1]], {n, 1, 20}]  (* Peter Luschny, Jan 04 2018 *)

a(n) = 2*sum(k=0, (n+1)\2, (4^k-1)*abs(stirling(n+1, 2*k, 1))*bernfrac(2*k)); \\ Seiichi Manyama, Oct 05 2022

a(n) = my(A = 1, B = 1); for(k=1, n, B *= k; A = (n-k+1)*A + B); A-B \\ Mikhail Kurkov, Aug 09 2025

def aList(n) -> list[int]:
    f, al, A = 1, 1, [1]
    for i in range(2, n + 1):
        f, al = f * i, (al + f) * (i + 1) >> 1
        A.append(al)
    return A
print(aList(22))  # Peter Luschny, Aug 09 2025

a(n) = A003149(n) - n!.
a(n) = A059371(n) + n! (n>=2).
a(n) = Sum_{k=1..n} k*A059438(n,k).
a(n) = Sum_{i=0..n-1} i!*(n-i)!.
a(n) = (n+1)!*(1 + Sum_{j=1..n-1} 2^j/(j+1))/2^n.
a(n) = (n+1)*a(n-1)/2 + (n-1)!*(n+1)/2, a(1)=1.
G.f.: f(f-1), where f(x) = Sum_{j>=0} j!*x^j.
a(n) = (n + 1)!*Re(-LerchPhi(2, 1, n + 1)). - Peter Luschny, Jan 04 2018
D-finite with recurrence: 2*a(n) +(-3*n+1)*a(n-1) +(n^2-3*n+4)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
a(n) = 2 * Sum_{k=0..floor((n+1)/2)} (4^k-1) * |Stirling1(n+1,2*k)| * Bernoulli(2*k). - Seiichi Manyama, Oct 05 2022
E.g.f.: x/((2-x)*(1-x)) - 2*log(1-x)/((2-x)^2). - Vladimir Kruchinin, Nov 16 2022

A145878 Triangle read by rows: T(n,k) is the number of permutations of [n] having k strong fixed points (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 14, 6, 3, 0, 1, 77, 29, 9, 4, 0, 1, 497, 160, 45, 12, 5, 0, 1, 3676, 1031, 249, 62, 15, 6, 0, 1, 30677, 7590, 1603, 344, 80, 18, 7, 0, 1, 285335, 63006, 11751, 2214, 445, 99, 21, 8, 0, 1, 2928846, 583160, 97056, 16168, 2865, 552, 119, 24

Offset: 0

Emeric Deutsch, Oct 29 2008

A permutation p of {1,2,...,n} is said to have j as a strong fixed point (splitter) if p(k) < j for k < j and p(k) > j for k > j.
T(n,k) is also the number of permutation graphs on n vertices with exactly k distinct dominating sets of size one. See the link by Theresa Baren, et al. -Daniel A. McGinnis, Oct 16 2018
The values T(k+r,k) are given as a polynomial expression in k when r is fixed, and the polynomial expressions can be calculated recursively. See the link by Theresa Baren, et al. -Daniel A. McGinnis, Oct 19 2018

			T(5,3) = 4 because we have 1'2'3'54, 1'2'435', 1'324'5' and 213'4'5' (the strong fixed points are marked).
Triangle starts:
   1;
   0,  1;
   1,  0,  1;
   3,  2,  0,  1;
  14,  6,  3,  0,  1;
  77, 29,  9,  4,  0,  1;

Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.

Nathaniel Johnston, Table of n, a(n) for n = 0..5050
Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801.
FindStat - Combinatorial Statistic Finder, The number of strong fixed points
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

Row sums gives A000142.

Cf. A052861, A006932, A003149.

n:=7: sfix:=proc(p) local ct,i: ct:= 0: for i to nops(p) do if p[i]=i and `subset`({seq(p[j],j=1..i-1)},{seq(k,k=1..i-1)})=true then ct:=ct+1 else end if end do: ct end proc: with(combinat): P:=permute(n): s:=[seq(sfix(P[j]),j= 1..factorial(n))]: for i from 0 to n do a[i]:=0 end do: for j to factorial(n) do if s[j]=0 then a[0]:=a[0]+1 elif s[j]=1 then a[1]:=a[1]+1 elif s[j]=2 then a[2]:=a[2]+1 elif s[j]=3 then a[3]:=a[3]+1 elif s[j]=4 then a[4]:=a[4]+1 elif s[j]=5 then a[5]:=a[5]+1 elif s[j]=6 then a[6]:=a[6]+1 elif s[j]=7 then a[7]:= a[7]+1 elif s[j]=8 then a[8]:=a[8]+1 elif s[j]=9 then a[9]:=a[9]+1 elif s[j]= 10 then a[10]:=a[10]+1 end if end do: seq(a[k],k=0..n); # yields row m of the triangle, where m is the value of n specified at the beginning of the program
n:=7: G:=1:for r from n to 2 by -1 do G:=1-(2*r-1)*z-(r^2*z^2)/G:od:G:=1/(1-t*z-z^2/G):
Gser := simplify(series(G, z = 0, n+1)): for m from 0 to n do seq(coeff(coeff(Gser, z, m), t, k), k = 0 .. m) end do; # based on P. Barry's g.f.; yields sequence in triangular form

nn=10;p=Sum[n!x^n,{n,0,nn}];i=1-1/p;CoefficientList[Series[1/(1-(i-x+y x)),{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Apr 27 2012 *)

T(n,0) = A052186(n).
Sum_{k=1..n} T(n,k) = A006932(n).
Sum_{k=0..n} k*T(n,k) = A003149(n-1).
G.f.: 1/(1-xy-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2/(1-... (continued fraction). - Paul Barry, Dec 09 2009
G.f.: 1/(1-(I(x)- x + y*x)) where I(x) is o.g.f. for A003319. - Geoffrey Critzer, Apr 27 2012
From Daniel A. McGinnis, Oct 15 2018: (Start)
T(n,k) = Sum_{i=1..n-k+1} T(n-i,k-1)*T(i-1,0).
T(3+k,k)=3k+3, T(4+k,k)=(k+1)(k+28)/2, T(5+k,k)=(k+1)(3k+77), T(6+k,k)=(k+1)(k^2+110k+2982)/6, T(7+k,k)=(k+1)(3k^2+235k+7352)/2 (previous conjectures).
See the link by Theresa Baren, et al. (End)

A309619 a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

Original entry on oeis.org

1, 1, 3, 7, 28, 128, 754, 5178, 41124, 368220, 3670872, 40290744, 482716896, 6267697920, 87664818960, 1313983544400, 21010949076960, 357007805477280, 6423473819220480, 122003441554176000, 2439346762501367040, 51213306647556506880, 1126446562222595147520

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A000142, A003149, A096161, A107713, A309618, A339515.

nmax = 25; CoefficientList[Series[Sum[k!*x^k, {k, 0, nmax}]*Sum[k!*x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Sum[k!*(n-2*k)!, {k, 0, Floor[n/2]}], {n, 0, 25}]

a(n) = sum(k=0, n\2, k! * (n - 2*k)!); \\ Michel Marcus, Dec 08 2020

G.f.: B(x)*B(x^2), where B(x) is g.f. of A000142.

a(n) ~ n! * (1 + 1/n^2 + 1/n^3 + 3/n^4 + 13/n^5 + 57/n^6 + 271/n^7 + 1467/n^8 + 8905/n^9 + 58965/n^10 + ...), for coefficients see A326984.

A324495 Average number of steps t(n) required to get n by repeatedly toggling one of the ceiling(log_2(n)) bits of the binary result of the previous step at a random position with equal probability of the bit positions, starting with all bits 0. The fractional part of t is given separately, i.e., t(n) = a(n) + A324496(n)/A324497(n).

Original entry on oeis.org

1, 3, 4, 7, 9, 9, 10, 15, 18, 18, 20, 18, 20, 20, 21, 31, 37, 37, 40, 37, 40, 40, 41, 37, 40, 40, 41, 40, 41, 41, 42, 63, 74, 74, 78, 74, 78, 78, 80, 74, 78, 78, 80, 78, 80, 80, 82, 74, 78, 78, 80, 78, 80, 80, 82, 78, 80, 80, 82, 80, 82, 82, 83, 127, 147, 147, 153

Offset: 1

Hugo Pfoertner, Mar 05 2019

nonn

The problem is related to random walks on the edges of n-dimensional hypercubes.

a(n) is only dependent on the length of the binary representation A070939(n) and on the binary weight A000120(n).

			a(5) = 9 is given by the sum of occurrence probabilities of toggle chains of even lengths 2*k, multiplied by the lengths.
a(5) = Sum_{k>=1} 4*k*7^(k-1) / 3^(2*k) = 9.
The corresponding simulation results for 10^10 toggle chains are
  2*k Probability P    2*k*P      Cumulated
    2   0.22222334  0.44444668    0.444447
    4   0.17284183  0.69136731    1.135814
    6   0.13442963  0.80657780    1.942392
    8   0.10455718  0.83645746    2.778849
   10   0.08131600  0.81315998    3.592009
   ...
  196   0.00000000  0.00000002    9.000068
.
a(7) = Sum_{k>=1} 2*(2*k+1)*7^(k-1) / 3^(2*k) = 10.

Hugo Pfoertner, Table of n, a(n) for n = 1..4096
P. Diaconis, R. L. Graham, J. A. Morrison, Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions, Technical Report EFS NFS 307, Department of Statistics, Stanford University, December 1988.
Persi Diaconis, R. L. Graham, J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions", Random Structures & Algorithms, Volume 1, Issue 1, Pages 51-72, Spring 1990.
IBM Research, Ponder This April 2006 - Challenge, Random walks on an n-dimensional hypercube.
IBM Research, Ponder This April 2006 - Challenge, Solution.

Cf. A000120, A003149, A070939, A099627, A324496, A324497.

A324496 Numerator of fractional part of average number of bit toggles to reach n as defined in A324495. The denominator is given in A324497.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 0, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 2, 3, 3, 4, 3, 4, 4, 1, 3, 4, 4, 1, 4, 1, 1, 1, 0, 0, 0, 3, 0, 3, 3, 4, 0, 3, 3, 4, 3, 4, 4, 11, 0, 3, 3, 4, 3, 4

Offset: 1

Hugo Pfoertner, Mar 05 2019

Hugo Pfoertner, Table of n, a(n) for n = 1..4096 (first 511 terms from Rainer Rosenthal)

Cf. A003149, A324495, A324497.

a(33)-a(64) from Rainer Rosenthal, Mar 07 2019

More terms from Rainer Rosenthal, Mar 19 2019

A324497 Denominator of fractional part of average number of bit toggles to reach n as defined in A324495. The numerator is given in A324496.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 6, 2, 6, 6, 3, 2, 6, 6, 3, 6, 3, 3, 3, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 5, 1, 5, 5, 5, 1, 5, 5, 5, 5, 5, 5, 15, 1, 5, 5, 5, 5, 5

Offset: 1

Hugo Pfoertner, Mar 05 2019

Hugo Pfoertner, Table of n, a(n) for n = 1..4096 (first 511 terms from Rainer Rosenthal)

Cf. A003149, A324495, A324496.

a(33)-a(64) from Rainer Rosenthal, Mar 07 2019

More terms from Rainer Rosenthal, Mar 19 2019

A024419 a(n) = n! (1/C(n,0) + 1/C(n,1) + ... + 1/C(n,[ n/2 ])).

Original entry on oeis.org

1, 1, 3, 8, 34, 156, 924, 6144, 48096, 420480, 4134240, 44720640, 530444160, 6824805120, 94787884800, 1412038656000, 22464536371200, 380017225728000, 6811416338227200, 128936055177216000, 2570286167543808000, 53818546503794688000, 1180914445357903872000

Offset: 0

Clark Kimberling

Half-convolution of factorials (A000142) with itself. For the definition of the half-convolution of a sequence with itself see a comment to A201204. - Vladimir Reshetnikov, Oct 05 2016

			a(3)=3!*(1/1 + 1/3)=6*4/3=8.

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A000142, A003149, A201204.

a:=proc(n) options operator, arrow: factorial(n)*(sum(1/binomial(n, k), k= 0.. floor((1/2)*n))) end proc: seq(a(n), n=0..21); # Emeric Deutsch, Oct 11 2007

Table[Sum[k! (n - k)!, {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 05 2016 *)

a(n) = sum(k=0, n\2, k!*(n-k)!); \\ Michel Marcus, Oct 05 2016

G.f.: (G(x)^2+H(x))/2 where G(x) = Sum_{k>=0} k!*x^k and H(x) = Sum_{k>=0} k!^2*x^(2*k). - Vladeta Jovovic, Sep 22 2007

a(n) = Sum_{k=0..floor(n/2)} k!*(n-k)!. - Vladimir Reshetnikov, Oct 05 2016

More terms from Emeric Deutsch, Oct 11 2007

A090595 Fourth column (k=3) of triangle A084938.

Original entry on oeis.org

1, 3, 9, 31, 126, 606, 3428, 22572, 170856, 1467432, 14123808, 150644448, 1763377344, 22466496960, 309371685120, 4577183527680, 72390548206080, 1218507923427840, 21746087150745600, 410094720409651200

Offset: 0

Philippe Deléham, Feb 01 2004

3rd column (k=2): A003149.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

B:=Binomial;; List([0..20], n-> Sum([0..n], k-> Sum([0..k], j-> Factorial(n)/(B(n,k)*B(k,j)) ))); # G. C. Greubel, Dec 29 2019

F:=Factorial; B:=Binomial; [ (&+[ (&+[F(n)/(B(k,j)*B(n,k)): j in [0..k]]) : k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

seq(factorial(n+2)*add(add(Beta(k+2, n-k+1)*Beta(j+1, k-j+1), j=0..k), k=0..n), n = 0..20); # G. C. Greubel, Dec 29 2019

Table[(n+2)!*Sum[Beta[k+2, n-k+1]*Beta[j+1, k-j+1], {k,0,n}, {j,0,k}], {n,0,20}] (* G. C. Greubel, Dec 29 2019 *)

vector(21, n, my(b=binomial); sum(k=0,n-1, sum(j=0,k, (n-1)!/(b(k,j)* b(n-1, k)) ))) \\ G. C. Greubel, Dec 29 2019

[ factorial(n+2)*sum(sum(beta(k+2,n-k+1)*beta(j+1,k-j+1) for j in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = Sum_{k=0..n} A003149(k)*(n-k)!.
G.f.: (Sum_{k>=0} k!*x^k)^3.
a(n) ~ 3 * n!. - Vaclav Kotesovec, Jun 25 2019
From G. C. Greubel, Dec 29 2019: (Start)
a(n) = (n+2)!*Sum_{k=0..n} Sum_{j=0..n} B(k+2, n-k+1)*B(j+1,k-j+1), where B(x,y) is the Beta function.
a(n) = Sum_{k=0..n} Sum_{j=0..k} n!/(binomial(n,k)*binomial(k,j)). (End)

cp's OEIS Frontend

A046826 Denominator of Sum_{k=0..n} 1/binomial(n,k).

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A126674 a(n) = n!*Sum_{j=0..n-1} 2^j/(j+1).

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A136128 Number of components in all permutations of [1,2,...,n].

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A145878 Triangle read by rows: T(n,k) is the number of permutations of [n] having k strong fixed points (0 <= k <= n).

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A309619 a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

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A324496 Numerator of fractional part of average number of bit toggles to reach n as defined in A324495. The denominator is given in A324497.

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A324497 Denominator of fractional part of average number of bit toggles to reach n as defined in A324495. The numerator is given in A324496.