A026725 -id:A026725 - cp's OEIS frontend

A026726 a(n) = T(2n,n), T given by A026725.

1, 2, 7, 27, 108, 440, 1812, 7514, 31307, 130883, 548547, 2303413, 9686617, 40783083, 171868037, 724837891, 3058850316, 12915186640, 54554594416, 230526280814, 974414815782, 4119854160332, 17422801069670, 73695109608352, 311768697325788, 1319136935150530

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Cf. A000045, A000108, A026725.

List([0..30], n-> Sum([0..n], k-> (2*k+1)*Binomial(2*n,n-k)*
Fibonacci(k+1)/(n+k+1) )); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 4*x*(1-Sqrt(1-4*x))/(8*x^2-(1-Sqrt(1-4*x))^3) )); // G. C. Greubel, Jul 16 2019

A026726 := proc(n)
    A026725(2*n,n) ;
end proc:
seq(A026726(n),n=0..10) ; # R. J. Mathar, Oct 26 2019

CoefficientList[Series[4*x*(1-Sqrt[1-4*x])/(8*x^2-(1-Sqrt[1-4*x])^3), {x,0,30}], x] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec(4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)) \\ G. C. Greubel, Jul 16 2019

(4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

From Philippe Deléham, Feb 11 2009: (Start)
a(n) = Sum_{k=0..n} A039599(n,k)*A000045(k+1).
a(n) = Sum_{k=0..n} A106566(n,k)*A122367(k). (End)
From Philippe Deléham, Feb 02 2014: (Start)
a(n) = Sum_{k=0..n} A236843(n+k,2*k).
a(n) = Sum_{k=0..n} A236830(n,k).
a(n) = A236830(n+1,1).
a(n) = A165407(3*n).
G.f.: C(x)/(1-x*C(x)^3), C(x) the g.f. of A000108. (End)
n*(5*n-11)*a(n) +2*(-20*n^2+59*n-30)*a(n-1) +15*(5*n^2-19*n+16)*a(n-2) +2*(5*n-6)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Oct 26 2019
n*a(n) +(-7*n+4)*a(n-1) +(7*n-2)*a(n-2) +(19*n-60)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Oct 26 2019

A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725.

Original entry on oeis.org

1, 4, 16, 65, 267, 1105, 4597, 19196, 80380, 337284, 1417582, 5965622, 25130844, 105954110, 447015744, 1886996681, 7969339643, 33670068133, 142301618265, 601586916703, 2543852427847, 10759094481491, 45513214057191, 192560373660245, 814807864164497

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Rob Arthan, Comments on A026674, A026725, A026670
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Also a(n) = T(2n-1, n-1), T given by A026670.

List([1..30], n-> Sum([1..n], k-> Binomial(2*n, n+k)*Fibonacci(k+1) *(k/n) )); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (-1+5*x +(1-x)*Sqrt(1-4*x))/(2*(1-4*x-x^2)) )); // G. C. Greubel, Jul 16 2019

a := n -> add(binomial(2*n,n+k)*combinat:-fibonacci(1+k)*(k/n), k=1..n):
seq(a(n), n=1..30); # Peter Luschny, Apr 28 2016

a[n_] := Sum[Binomial[2n, n+k] Fibonacci[k+1] k/n, {k, 1, n}];
Array[a, 30] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)

a(n):=sum(k*binomial(2*n,n-k)*(sum(binomial(k-i,i),i,0,k/2)),k,1,n)/n; /* Vladimir Kruchinin, Apr 28 2016 */

a(n)=sum(k=1,n,k*binomial(2*n,n-k)*sum(i=0,k\2,binomial(k-i,i)))/n \\ Charles R Greathouse IV, Apr 28 2016

a=((-1+5*x +(1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 16 2019

G.f.: (1/2)*((1-x)/(sqrt(1-4*x)-x) - 1). - Ralf Stephan, Feb 05 2004
G.f.: x*c(x)^3/(1-x*c(x)^3) = (1-5*x -(1-x)*sqrt(1-4*x))/(2*(x^2+4*x-1)), c(x) the g.f. of A000108. - Paul Barry, Mar 19 2007
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is the upper left term in M^n, where M is the following infinite square production matrix:
   1, 1, 0, 0, 0, 0, 0, ...
   3, 1, 1, 0, 0, 0, 0, ...
   6, 1, 1, 1, 0, 0, 0, ...
  10, 1, 1, 1, 1, 0, 0, ...
  15, 1, 1, 1, 1, 1, 0, ...
  21, 1, 1, 1, 1, 1, 1, ...
  ... (End)
D-finite with recurrence n*a(n) +(-9*n+8)*a(n-1) +23*(n-2)*a(n-2) +(-11*n+48)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
a(n) = (1/n)*Sum_{k=1..n} k*binomial(2*n,n-k)*Sum_{i=0..k/2} binomial(k-i,i). - Vladimir Kruchinin, Apr 28 2016
a(n) ~ (3 - sqrt(5)) * (2 + sqrt(5))^n / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

A026842 a(n) = T(2n,n-3), T given by A026725.

Original entry on oeis.org

1, 9, 56, 300, 1487, 7041, 32381, 146017, 649395, 2859231, 12494914, 54291912, 234860677, 1012433965, 4352210327, 18666918033, 79916230409, 341615895659, 1458457275715, 6220016154525, 26503542364381, 112847001503099, 480173686483581

Offset: 3

Clark Kimberling

nonn

Column k=8 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A236830, A026846, A026849.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^8/(32*x^3*(8*x^2 -(1 - Sqrt[1-4*x])^3 )), {x,0,30}], x],3] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1 - sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 30).coefficients(x, sparse=False); a[3:] # G. C. Greubel, Jul 17 2019

a(n) = A026846(n) = A026849(n). - Philippe Deléham, Feb 02 2014
G.f.: (x^3*C(x)^8)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014
D-finite with recurrence -(n+3)*(253*n-940)*a(n) +(3061*n^2-6571*n-18156)*a(n-1) +(-12091*n^2+43849*n-996)*a(n-2) +(14543*n^2-76721*n+109596)*a(n-3) +2*(1037*n-2568)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 22 2025

A026846 a(n) = T(2n+1,n+4), T given by A026725.

Original entry on oeis.org

1, 9, 56, 300, 1487, 7041, 32381, 146017, 649395, 2859231, 12494914, 54291912, 234860677, 1012433965, 4352210327, 18666918033, 79916230409, 341615895659, 1458457275715, 6220016154525, 26503542364381, 112847001503099, 480173686483581

Offset: 3

Clark Kimberling

nonn

This is probably the same as A026842 because A026725 is built in a left-right symmetric Pascal-tree-summation fashion. - R. J. Mathar, May 28 2008

Column k=8 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A236830, A026842, A026846.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-Sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^8/(32*x^3*(8*x^2 -(1 - Sqrt[1-4*x])^3 )), {x,0,30}], x], 3] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1 - sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=((1-sqrt(1-4*x))^8/(32*x^3*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 30).coefficients(x, sparse=False); a[3:] # G. C. Greubel, Jul 17 2019

a(n) = A026842(n) = A026849(n). - Philippe Deléham, Feb 02 2014

G.f.: (x^3*C(x)^8)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

A026732 a(n) = Sum_{k=0..n} T(n,k), T given by A026725.

Original entry on oeis.org

1, 2, 4, 9, 18, 40, 80, 176, 352, 769, 1538, 3343, 6686, 14477, 28954, 62505, 125010, 269216, 538432, 1157244, 2314488, 4966260, 9932520, 21282622, 42565244, 91096110, 182192220, 389515284, 779030568, 1664015246, 3328030492

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

T:= function(n,k)
    if k=0 or k=n then return 1;
    elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
    else return T(n-1, k-1) + T(n-1, k);
    fi;
  end;
List([0..30], n-> Sum([0..n], k-> T(n,k) )); # G. C. Greubel, Oct 26 2019

A026732 := proc(n)
    add(A026725(n,k),k=0..n) ;
end proc:
seq(A026732(n),n=0..10) ; # R. J. Mathar, Oct 26 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Oct 26 2019 *)

T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(31, n, sum(k=0,n-1, T(n-1,k)) ) \\ G. C. Greubel, Oct 26 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Oct 26 2019

Conjecture: +(-n+1)*a(n) +2*a(n-1) +3*(3*n-7)*a(n-2) -10*a(n-3) +(-23*n+95)*a(n-4) +6*a(n-5) +(11*n-95)*a(n-6) +2*a(n-7) +4*(n-7)*a(n-8)=0. - R. J. Mathar, Oct 26 2019

A026841 a(n) = T(2n,n-4), T given by A026725.

Original entry on oeis.org

1, 11, 79, 471, 2535, 12809, 62067, 292085, 1345718, 6102780, 27343148, 121359692, 534632836, 2341151646, 10201950700, 44278673806, 191540714294, 826265471868, 3555992623850, 15273547250820, 65491352071266, 280412963707416

Offset: 4

Clark Kimberling

nonn

Column k=10 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 4..1000

Cf. A236830.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1-Sqrt(1-4*x))^3 )) )); // G. C. Greubel, Jul 17 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^10/(128*x^4*(8*x^2 -(1 - Sqrt[1-4*x])^3 )), {x,0,40}], x], 4] (* G. C. Greubel, Jul 17 2019 *)

my(x='x+O('x^40)); Vec((1-sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1 - sqrt(1-4*x))^3 ))) \\ G. C. Greubel, Jul 17 2019

a=((1-sqrt(1-4*x))^10/(128*x^4*(8*x^2 -(1-sqrt(1-4*x))^3 ))).series(x, 45).coefficients(x, sparse=False); a[4:40] # G. C. Greubel, Jul 17 2019

a(n) = A026848(n). - Philippe Deléham, Feb 02 2014
G.f.: (x^4*C(x)^10)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014
D-finite with recurrence (n+4)*(3421*n+2687)*a(n) +(3421*n^2-245139*n-819238)*a(n-1) +3*(-126201*n^2+820641*n+1451992)*a(n-2) +(1944367*n^2-12105285*n+5094446)*a(n-3) +6*(-438489*n^2+3204217*n-6453730)*a(n-4) -12*(2*n-7)*(30601*n-111490)*a(n-5)=0. - R. J. Mathar, Jul 22 2025

A026672 a(n) = T(2n,n-1), T given by A026670. Also T(2n,n-1)=T(2n+1,n+2), T given by A026725; and T(2n,n-1), T given by A026736.

Original entry on oeis.org

1, 5, 22, 94, 398, 1680, 7085, 29877, 126021, 531751, 2244627, 9478605, 40040183, 169193597, 715143046, 3023492646, 12785541850, 54076955716, 228759017624, 967850695362, 4095387893312, 17331318506030

Offset: 2

Clark Kimberling

nonn

Column k=4 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A000108, A026670, A026725, A026736, A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^4/(2*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^4/(2*(8*x^2 -(1-Sqrt[1-4*x] )^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^4/(2*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019

a=((1-sqrt(1-4*x))^4/(2*(8*x^2 -(1-sqrt(1-4*x))^3))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 16 2019

G.f.: (x*C(x)^4)/(1-x*C(x)^3), where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

Conjecture: -(n+1)*(n-6)*a(n) +2*(4*n^2-23*n+3)*a(n-1) +3*(-5*n^2+33*n-42)*a(n-2) -2*(2*n-3)*(n-5)*a(n-3)=0. - R. J. Mathar, Aug 08 2015

A026675 a(n) = T(2n-1,n-2), T given by A026670. Also T(2n-1,n-2) = T(2n,n+2), T given by A026725 and T(2n,n-2), T given by A026736.

Original entry on oeis.org

1, 6, 29, 131, 575, 2488, 10681, 45641, 194467, 827045, 3512983, 14909339, 63239487, 268127302, 1136495965, 4816202207, 20406887583, 86457399359, 366263778659, 1551535465465, 6572224024539, 27838835937511, 117918419518219

Offset: 2

Clark Kimberling

nonn

Column k=5 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A000108, A026670, A026725, A026736, A236830.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(  (1-Sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 16 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^5/(4*x*(8*x^2 -(1-Sqrt[1 - 4*x])^3)), {x,0,30}], x], 2] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec( (1-sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-sqrt(1-4*x))^3))) \\ G. C. Greubel, Jul 16 2019

a=((1-sqrt(1-4*x))^5/(4*x*(8*x^2 -(1-sqrt(1-4*x))^3))).series(x, 30).coefficients(x, sparse=False); a[2:] # G. C. Greubel, Jul 16 2019

G.f.: (x^2*C(x)^5)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

A026733 a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026725.

Original entry on oeis.org

1, 1, 3, 5, 13, 23, 57, 103, 249, 455, 1083, 1993, 4693, 8679, 20275, 37633, 87377, 162643, 375789, 701075, 1613413, 3015563, 6916957, 12948083, 29617161, 55513327, 126678893, 237705547, 541325021, 1016736115, 2311294377

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

A026733 := proc(n)
    add(A026725(n,k),k=0..floor(n/2)) ;
end proc:
seq(A026733(n),n=0..10) ; # R. J. Mathar, Oct 26 2019

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]]; Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Oct 26 2019 *)

T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
vector(31, n, sum(k=0,floor(n-1/2), T(n-1,k)) ) \\ G. C. Greubel, Oct 26 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==1 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Oct 26 2019

Conjecture: (-n+2)*a(n) +(n-2)*a(n-1) +2*(4*n-13)*a(n-2) +8*(-n+4)*a(n-3) +5*(-3*n+14)*a(n-4) +(15*n-94)*a(n-5) +2*(-2*n+9)*a(n-6) +4*(n-6)*a(n-7)=0. - R. J. Mathar, Oct 26 2019

A026843 a(n) = T(2n,n+3), T given by A026725.

Original entry on oeis.org

1, 8, 46, 233, 1108, 5083, 22805, 100827, 441311, 1917751, 8289965, 35694218, 153225617, 656213596, 2805143526, 11973556060, 51047361676, 217420991444, 925300665762, 3935293406942, 16727533586006, 71069911887898, 301835332909216

Offset: 3

Clark Kimberling

nonn

Column k=7 of triangle A236830. - Philippe Deléham, Feb 02 2014

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A000108, A026725, A236830.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))^7/(16*x^2*(8*x^2 -(1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 19 2019

Drop[CoefficientList[Series[(1-Sqrt[1-4*x])^7/(16*x^2*(8*x^2 -(1-Sqrt[1-4*x])^3)), {x, 0, 40}], x], 3] (* G. C. Greubel, Jul 19 2019 *)

my(x='x+O('x^40)); Vec( (1-sqrt(1-4*x))^7/(16*x^2*(8*x^2 -(1-sqrt(1-4*x))^3)) ) \\ G. C. Greubel, Jul 19 2019

a=((1-sqrt(1-4*x))^7/(16*x^2*(8*x^2 -(1-sqrt(1-4*x))^3)) ).series(x, 45).coefficients(x, sparse=False); a[3:40] # G. C. Greubel, Jul 19 2019

G.f.: (x^3*C(x)^7)/(1-x*C(x)^3) where C(x) is the g.f. of A000108. - Philippe Deléham, Feb 02 2014

a(n) ~ phi^(3*n-4) / sqrt(5), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 19 2019

cp's OEIS Frontend

A026726 a(n) = T(2n,n), T given by A026725.

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A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725.

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A026842 a(n) = T(2n,n-3), T given by A026725.

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Magma

Mathematica

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A026846 a(n) = T(2n+1,n+4), T given by A026725.

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A026732 a(n) = Sum_{k=0..n} T(n,k), T given by A026725.

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A026841 a(n) = T(2n,n-4), T given by A026725.

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Magma

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