A055113 -id:A055113 - cp's OEIS frontend

A185286 Triangle T(n,k) is the number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at k.

1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 5, 6, 3, 3, 3, 1, 11, 11, 13, 17, 13, 7, 6, 4, 1, 24, 41, 52, 44, 43, 40, 25, 14, 10, 5, 1, 93, 120, 152, 176, 161, 126, 107, 80, 45, 25, 15, 6, 1, 272, 421, 550, 559, 561, 524, 412, 303, 227, 146, 77, 41, 21, 7, 1, 971, 1381, 1813, 2056, 2045, 1835, 1615, 1309, 938, 648, 435, 251, 126, 63, 28, 8, 1

Offset: 0

Franklin T. Adams-Watters, Mar 10 2011

Equivalently, the number of paths from (0,0) to (n,k) using steps of the form (1,2),(1,1),(1,-1) or (1,-2) and staying on or above the x-axis.

It appears that A047002 gives the row sums of this triangle.

			The table starts:
1
0,1,1
2,1,1,2,1
2,5,6,3,3,3,1

Robert Israel, Table of n, a(n) for n = 0..10200

Columns k=0..2 are A187430, A055113, A296619.

Cf. A005408(row lengths), A047002(apparently row sums).

T:= proc(n,k) option remember;
  if k < 0 or k > 2*n then return 0 fi;
  procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k+1)+procname(n-1,k+2)
end proc:
T(0,0):= 1:
for nn from 0 to 10 do
  seq(T(nn,k),k=0..2*nn)
od; # Robert Israel, Dec 19 2017

T[n_, k_] := T[n, k] = If[k < 0 || k > 2n, 0, T[n-1, k-2] + T[n-1, k-1] + T[n-1, k+1] + T[n-1, k+2]];
T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Aug 19 2022, after Robert Israel *)

flip(v)=vector(#v,i,v[#v+1-i])
ar(n)={local(p);p=1;
for(k=1,n,p*=1+x+x^3+x^4;p=(p-polcoeff(p,0)-polcoeff(p,1)*x)/x^2);
flip(Vec(p))}

A278618 a(n) = Sum_{j=0..n/2} binomial(n-j-1,n-2j)binomial(2*n+1,j).

Original entry on oeis.org

1, 0, 5, 7, 45, 121, 533, 1800, 7157, 26239, 101640, 384583, 1483925, 5693247, 22013059, 85076183, 330014421, 1281349195, 4985766650, 19422653367, 75775163028, 295953650376, 1157212653030, 4529183513913, 17743019073381, 69565441895001

Offset: 0

Vladimir Kruchinin, Nov 23 2016

nonn

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

Cf. A005043, A055113.

Table[Sum[Binomial[n - j - 1, n - 2*j]*Binomial[2*n + 1, j], {j, 0, n/2}], {n,0,50}] (* G. C. Greubel, Jun 06 2017 *)

A(x):=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x));
B(x):=1/((x+1)*sqrt(-3*x^2-2*x+1));
C(x):=sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4;
taylor(x*A(x)/C(x)*B(C(x)),x,0,20);

for(n=0,25, print1(sum(j=0,n, binomial(n-j-1,n-2*j)*binomial(2*n+1,j)), ", ")) \\ G. C. Greubel, Jun 06 2017

G.f.: x*A(x)/C(x)*B(C(x)), where
A(x) = (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)),
B(x) = 1/((x+1)*sqrt(-3*x^2-2*x+1)),
C(x) = sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4.
a(n) ~ (1 - 1/sqrt(5)) * 4^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 24 2016
a(n) = (2*n + 1)*3F2(1-n/2,3/2-n/2,-2*n; 2,2-n; 4) for n>1. - Ilya Gutkovskiy, Nov 24 2016
Conjecture: 2*n*(5*n-8)*(2*n-1)*(n+1)*a(n) -n*(115*n^3-344*n^2+299*n-82)*a(n-1) -4*(2*n-1)*(5*n^3+27*n^2-74*n+30)*a(n-2) +36*(n-2)*(5*n-3)*(2*n-1)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2016

A296619 The number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 2.

Original entry on oeis.org

0, 1, 1, 6, 13, 52, 152, 550, 1813, 6453, 22427, 80330, 286895, 1038931, 3772801, 13807294, 50726893, 187332517, 694364517, 2583714636, 9644852364, 36115537269, 135607526865, 510496492338, 1926284451923, 7284476707597, 27602839227883, 104791979218326

Offset: 0

Feng Jishe, Dec 17 2017

a(n) is the number of 2-D walks with n steps of type {(1,-2), (1,-1), (1,1), or (1,2)} starting at (0,0), ending at (n,2), and not dropping below the x-axis.

The sequence corresponds to element (1,3) of the matrix B(n)^n (see Maple script). Furthermore, element (1,1) of the matrix is A187430, the element (1,2) of these matrix is A055113.

			There are 6 walks of length 3:
        __
       |  |         __
     __|  |_     __|  |_     __    _
    |           |           |  |__|
   _|          _|          _|
    2+2-2=2     2+1-1=2     2-1+1=2
                    __
     __    _       |  |_           _
    |  |  |      __|         __   |
   _|  |__|    _|          _|  |__|
    2-2+2=2     1+2-1=2     1-1+2=2

Robert Israel, Table of n, a(n) for n = 0..1667

Cf. A055113, A185286, A187430.

B := n -> LinearAlgebra:-ToeplitzMatrix([0,1,1, seq(0, k=0..n-2)], symmetric):
seq((B(n)^n)(1, 3), n=0..27);
# alternative:
T:= proc(n,k) option remember;
  if k < 0 or k > 2*n then return 0 fi;
  procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k+1)+procname(n-1,k+2)
end proc:
T(0,0):= 1:
seq(T(n,2),n=0..40); # Robert Israel, Dec 19 2017

b[n_] := ToeplitzMatrix[Join[{0,1,1}, ConstantArray[0,n-1]]];
Prepend[Table[MatrixPower[b[n],n][[1,3]], {n,20}], 0]
(* Andrey Zabolotskiy, Dec 19 2017 *)

Next(v)={vector(#v+2, i, if(i<3||i>#v-2, 0, v[i-2]+v[i-1]+v[i+1]+v[i+2]))}
my(v=vector(7,i,i==3)); for(n=1, 50, print1(v[5],", "); v=Next(v)) \\ Andrew Howroyd, Dec 18 2017

a(n) = A185286(n,2). - Robert Israel, Dec 19 2017

A123880 Inverse of number triangle A123878.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 5, 3, 5, 0, 1, 11, 18, 5, 7, 0, 1, 41, 39, 35, 7, 9, 0, 1, 120, 157, 75, 56, 9, 11, 0, 1, 421, 459, 325, 119, 81, 11, 13, 0, 1, 1381, 1668, 950, 553, 171, 110, 13, 15, 0, 1, 4840

Offset: 0

Paul Barry, Oct 16 2006

First column is A055113. Row sums are A000108.

			Number triangle begins
1,
0, 1,
1, 0, 1,
1, 3, 0, 1,
5, 3, 5, 0, 1,
11, 18, 5, 7, 0, 1,
41, 39, 35, 7, 9, 0, 1,
120, 157, 75, 56, 9, 11, 0, 1

A211867 a(n) = A097609(2*n-1,n), n>0; a(0)=1.

Original entry on oeis.org

1, 0, 2, 3, 18, 50, 215, 735, 2898, 10668, 41202, 156090, 601623, 2308878, 8923343, 34487453, 133749330, 519277512, 2020262660, 7869597840, 30699524018, 119894389380, 468768069882, 1834589752182, 7186572436887, 28175111736300, 110547143014050, 434049816801900

Offset: 0

Vladimir Kruchinin, Feb 12 2013

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1665
D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
Eric Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, arXiv:1709.07996 [math.CO], 2017.

a := n -> (-1)^n*binomial(2*n-1,n-1)*hypergeom([-n,n/2,(n+1)/2], [n,n+1], 4):
seq(simplify(a(n)), n=0..27); # Peter Luschny, Nov 02 2016

a[n_] := ((-1)^(3*n)*(2*n)!*HypergeometricPFQ[{(n+1)/2, -n, n/2}, {n, n+1}, 4])/(2*n!^2); a[0]=1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 13 2013, from A097609 *)

a(n) = if(n==0, 1, sum(k=0, n/2, (binomial(2*n,k)*binomial(n-k-1,n-2*k))/2)); \\ Altug Alkan, Oct 05 2015

G.f.: x*G'(x)/G(x), where G(x) is the g.f. of A055113.
G.f.: x * d/dx (log(sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4)).
a(n) = sum(j=0..n, C(2*j+n-1,j)*(-1)^(n+j)*C(2*n,n-j))/2, n>0; a(0)=1.
a(n) = A097609(2*n-1,n), n>0; a(0)=1. (Corrected by M. F. Hasler, Feb 12 2013)
a(n) = Sum_{j=0..n/2} (binomial(2*n,j)*binomial(n-j-1,n-2*j))/2. - Vladimir Kruchinin, Oct 05 2015
a(n) ~ 2^(2*n-1) / sqrt(5*Pi*n). - Vaclav Kotesovec, Apr 27 2024

A239425 Expansion of -16/(sqrt(12x+2sqrt(1-4x)+2)-sqrt(1-4x)-1)^2+1/x^2-1.

Original entry on oeis.org

1, 2, 7, 16, 53, 156, 522, 1702, 5833, 19990, 70079, 247160, 882587, 3172196, 11492847, 41874864, 153452521, 564975570, 2089346157, 7756501690, 28898156364, 108010059036, 404890987653, 1521877280868, 5734545323859, 21657665796526

Offset: 0

Vladimir Kruchinin, Mar 17 2014

nonn

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

Cf. A097609, A055113.

CoefficientList[Series[-16/(Sqrt[12*x+2*Sqrt[1-4*x]+2]-Sqrt[1-4*x] -1)^2+1/x^2-1, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 18 2014 *)
Flatten[{1,Table[Sum[Binomial[n+2*j-1,j+n-1]*(-1)^(j+n)*Binomial[2*n+2,j+n],{j,0,n+2}]/(n+1),{n,1,20}]}] (* Vaclav Kotesovec, Mar 18 2014 *)

a(n):=(sum(binomial(n+2*j-1, j)*(-1)^(j+n)*binomial(2*n+2, j+n), j, 0, n+2))/(n+1)-kron_delta(n,0);

my(x='x+O('x^50)); Vec(-16/(sqrt(12*x+2*sqrt(1-4*x)+2)-sqrt(1-4*x) -1)^2 + 1/x^2 -1) \\ G. C. Greubel, Jun 01 2017

a(n) = (Sum_{j=0..(n+2)} C(n+2*j-1,j)*(-1)^(j+n)*C(2*n+2,j+n))/(n+1) - delta(n,0).
a(n) ~ (5+3*sqrt(5)) * 2^(2*n+1) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 18 2014
Conjecture: 2*(2*n+1)*(n+2)*(n+1)*a(n) +(n+1)*(n^2-27*n+2)*a(n-1) +2*(-73*n^3+204*n^2-167*n+6)*a(n-2) +12*(n-3)*(2*n-3)*(4*n-7)*a(n-3) +216*(2*n-5)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Apr 02 2014

A212696 Central coefficient of the triangle A097609.

Original entry on oeis.org

1, 0, 3, 4, 25, 66, 287, 960, 3789, 13810, 53240, 200652, 771641, 2952054, 11386065, 43910288, 170007429, 658979586, 2560258550, 9960335060, 38811668868, 151418146704, 591464244882, 2312774560296, 9052560751725, 35464735083726, 139054217427702, 545635715465596

Offset: 0

Vladimir Kruchinin, May 24 2012

nonn

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Cf. A007318, A027907, A055113, A097609.

Table[((n + 1) Sum[Binomial[n + 2 j, n + j] (-1)^(n - j) Binomial[2 n + 1, n + j + 1], {j, 0, n}])/(2 n + 1), {n, 0, 27}] (* or *)
CoefficientList[Series[(12 - 4/#)/(8 Sqrt[12 x + 2 # + 2]) + 1/(2 #) &@ Sqrt[1 - 4 x], {x, 0, 27}], x] (* Michael De Vlieger, Oct 08 2016 *)
a[n_] := (-1)^n Binomial[2n, n] HypergeometricPFQ[{(n+1)/2, 1+n/2, -n}, {1+n, 2+n}, 4]; Table[a[n], {n, 0, 27}] (* Peter Luschny, Dec 26 2017 *)

x='x+O('x^66);
gf=(12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x));
Vec(Ser(gf))
/* Joerg Arndt, Jun 09 2012 */

G.f.: (12-4/sqrt(1-4*x))/(8*sqrt(12*x+2*sqrt(1-4*x)+2))+1/(2*sqrt(1-4*x)).
a(n) = ((n+1)*Sum_{j=0..n} C(n+2*j, n+j)*(-1)^(n-j)*C(2*n+1, n+j+1)) / (2*n+1).
a(n) = (n+1)*A055113(n).
Conjecture: 2*n*(n-1)*(2*n+1)*(5*n-8)*a(n) -(n-1)*(115*n^3-344*n^2+299*n-82) *a(n-1) -4*(2*n-3)*(5*n^3+27*n^2-74*n+30)*a(n-2) +36*(n-1)*(5*n-3)*(2*n-3)*(2*n-5) *a(n-3)=0. - R. J. Mathar, Oct 08 2016
a(n) = (-1)^n*binomial(2*n, n)*hypergeom([(n+1)/2, 1+n/2, -n], [1+n, 2+n], 4). - Peter Luschny, Dec 26 2017
From Emanuele Munarini, Jul 14 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k)*binomial(n-k-1,k-1)*(n+1)/(2*n-k+1).
a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,k)*binomial(3n-2k,2*n-k)*(n+1)/(2*n-k+1).
a(n) = (n+1)/(2n+1)*Sum_{k=0..n} binomial(2*n+i,2*n)*trinomial(2*n+1,n-k)*(-1)^{n-k}, where trinomial(n,k) are the trinomial coefficients (A027907).
a(n) = Sum_{k=0..n} (-1)^k*binomial(3*n-k,n-k)*trinomial(2*n,k)*(n+k+1)/(2*n+1). (End)

A239230 Expansion of -xlog'(-sqrt(12x+2sqrt(1-4x)+2)/4+sqrt(1-4*x)/4+5/4).

Original entry on oeis.org

0, 1, 1, 4, 9, 36, 112, 428, 1505, 5692, 21026, 79806, 301488, 1151866, 4403778, 16929474, 65204353, 251947668, 975366094, 3784197606, 14705937794, 57242631464, 223121176224, 870805992278, 3402485053664, 13308485156086, 52104519751272, 204176144516818

Offset: 0

Vladimir Kruchinin, May 22 2014

nonn

Cf. A055113.

a:= n-> add((-1)^(k+n)*binomial(2*n-k-1, n-1)*hypergeom([k-n, (n+1)/2, n/2], [n, n+1], 4), k=1..n);
seq(round(evalf(a(n), 32)), n=0..24); # Peter Luschny, May 22 2014

a(n):=n*sum(sum(binomial(n+2*j-1,j+n-1)*(-1)^(k+j+n)*binomial(2*n-k,j+n), j,0,n-k)/(2*n-k), k,1,n);

G.f.: A(x) = x*F'(x)/(1-F(x)), where F(x) is g.f. of A055113.
a(n) = n * Sum_{k=1..n} (Sum_{j=0..n-k} C(n+2*j-1,j+n-1) * (-1)^(k+j+n) * C(2*n-k,j+n)) / (2*n-k).
a(n) = Sum_{k=1..n} (-1)^(k+n) * C(2*n-k-1,n-1) * hypergeom([k-n, n/2+1/2, n/2], [n, n+1], 4). - Peter Luschny, May 22 2014
Conjecture D-finite with recurrence +24*(365025561*n-1672569283)*(n-1)*(n-2)*(2*n-1)*a(n) -4*(n-2)*(27129169947*n^3-209577621466*n^2+463278020461*n-314084557758)*a(n-1) +2*(-28823853823*n^4+487259692534*n^3-3105214937957*n^2+8814274338098*n-9143920331436)*a(n-2) +4*(276083065830*n^4-4172118623320*n^3+24824880820695*n^2-69263721795041*n+75832154222148)*a(n-3) +(-587491214125*n^4+9941738070620*n^3-75070680472775*n^2+281912285021344*n-413197788157152)*a(n-4) +2*(-1186924847911*n^4+26108844767699*n^3-211936472383904*n^2+757584729548632*n-1009721693733312)*a(n-5) +4*(2*n-11)*(328530544924*n^3-5280431217363*n^2+28334632524947*n-50473913356356)*a(n-6) -72*(4143100547*n-18456753180)*(n-6)*(2*n-11)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Jul 27 2022
Conjecture: a(n) = Sum_{i=0..n-1} A059260(2*(n-1),n+i-1)*A009766(n+i-1,i)*(-1)^(n+i-1) = n*Sum_{i=0..n-1} binomial(n+2*i, i)/(n+2*i)*(-1)^(n+i-1)*[x^(n+i-1)] (1+(x+1)^(2*n-1))/(x+2) for n >= 0. - Mikhail Kurkov, Feb 18 2025

A297074 Number of ways of inserting parentheses in x^x^...^x (with n x's) whose result is an integer where x = sqrt(2).

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 10, 23, 55

Offset: 1

Jon E. Schoenfield, Dec 24 2017

The largest value that can be obtained by inserting parentheses in x^x^x^x^x^x^x^x^x (9 x's), where x = sqrt(2), is x^(x^((((((x^x)^x)^x)^x)^x)^x)) = 2^128 = 340282366920938463463374607431768211456; this is one of the a(9) = 55 ways of inserting parentheses in x^x^x^x^x^x^x^x^x that yield an integer value.

			With x = sqrt(2),
x = 1.414213... is not an integer, so a(1) = 0;
x^x = 1.632526... is not an integer, so a(2) = 0.
(x^x)^x = 2 is an integer, but x^(x^x) = 1.760839... is not, so a(3) = 1;
((x^x)^x)^x, (x^x)^(x^x), (x^(x^x))^x, and x^(x^(x^x)) are noninteger values, but x^((x^x)^x) = 2, so a(4) = 1;
the only ways of inserting parentheses in x^x^x^x^x that yield integer values are x^(x^((x^x)^x)) = 2 and (((x^x)^x)^x)^x = 4, so a(5) = 2.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Index entries for sequences related to parenthesizing

Cf. A002845, A055113, A082499, A198683.

With[{x = Sqrt@ 2}, Array[Count[#, ?IntegerQ] &@ Map[ToExpression@ StringReplace[ToString@ #, {"{" -> "(", "}" -> ")", "," -> "^"}] &, Groupings[#, 2] /. _Integer -> x] &, 9]] (* _Michael De Vlieger, Dec 24 2017 *)

A346380 Complement of A187430 in A000108.

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 18, 39, 157, 459, 1668, 5503, 19638, 68325, 245144, 876438, 3177651, 11549939, 42307920, 155555733, 574881920, 2132231076, 7938771624, 29651189637, 111086480106, 417305224917, 1571633677078, 5932720163529, 22443721850064, 85075094996719, 323086777251300

Offset: 0

F. Chapoton, Jul 14 2021

nonn

Related to the decomposition of A000108 as the sum of A055113 and A111160.

Cf. A000108, A055113, A111160, A187430.

N = 30
x = (PowerSeriesRing(QQ, 'x').0).O(N + 1)
f = (x*(1-x^2)^2/(1+x^3)^2).reverse()
g = sum(catalan_number(n)*x**n for n in range(N + 1)).O(N + 1)
list(x*g-f)

G.f. y(x) satisfies x y^4 + 2 x^2 y^2 + x^3 + 3 x y^2 + y^3 - x y = 0.

cp's OEIS Frontend

A185286 Triangle T(n,k) is the number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at k.

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A278618 a(n) = Sum_{j=0..n/2} binomial(n-j-1,n-2*j)*binomial(2*n+1,j).

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A296619 The number of nonnegative walks of n steps with step sizes 1 and 2, starting at 0 and ending at 2.

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A123880 Inverse of number triangle A123878.

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A211867 a(n) = A097609(2*n-1,n), n>0; a(0)=1.

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A239425 Expansion of -16/(sqrt(12*x+2*sqrt(1-4*x)+2)-sqrt(1-4*x)-1)^2+1/x^2-1.

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Mathematica

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A212696 Central coefficient of the triangle A097609.

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A239230 Expansion of -x*log'(-sqrt(12*x+2*sqrt(1-4*x)+2)/4+sqrt(1-4*x)/4+5/4).

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Programs

A278618 a(n) = Sum_{j=0..n/2} binomial(n-j-1,n-2j)binomial(2*n+1,j).

A239425 Expansion of -16/(sqrt(12x+2sqrt(1-4x)+2)-sqrt(1-4x)-1)^2+1/x^2-1.

A239230 Expansion of -xlog'(-sqrt(12x+2sqrt(1-4x)+2)/4+sqrt(1-4*x)/4+5/4).