cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A214692 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 2, 11, 71, 515, 3997, 32488, 273009, 2352724, 20678966, 184660333, 1670619561, 15279692008, 141048655988, 1312429249996, 12296515232446, 115909188223053, 1098444610424929, 10459429664510189, 100021237512559055, 960168745226226195, 9249466125601138425
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 71*x^4 + 515*x^5 + 3997*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^5 + (A(x)-1)*(A(x)^3-1)/A(x)^12 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^21 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^32 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^45 +...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 26*x^3 + 168*x^4 + 1216*x^5 + 9429*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 46*x^3 + 297*x^4 + 2148*x^5 + 16649*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 465*x^4 + 3364*x^5 + 26078*x^6 +...
where 1+x = A(x)^2 + A(x)^3 - A(x)^4.

Links

Crossrefs

Cf. A214690, A214693, A214694, A214695, A001002 (variant).

Programs

  • Mathematica
    CoefficientList[1+InverseSeries[Series[x - 2*x^2 - 3*x^3 - x^4, {x, 0, 20}], x],x] (* Vaclav Kotesovec, Nov 29 2014 *)
  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 2*x^2 - 3*x^3 - x^4 +x^2*O(x^n)), n))}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(4*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* From 1+x = A(x)^2 + A(x)^3 - A(x)^4: */
{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec(Ser(A)^2+Ser(A)^3-Ser(A)^4)[#A]);A[n+1]}
for(n=0,25,print1(a(n) ,", "))

Formula

G.f. A(x) satisfies:
(1) 1+x = A(y) where y = x - 2*x^2 - 3*x^3 - x^4, which is the g.f. of row 2 in triangle A214690.
(2) x = Sum_{n>=1} 1/A(x)^(n*(n+4)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
(3) 1+x = A(x)^2 + A(x)^3 - A(x)^4. - Paul D. Hanna, Nov 15 2014
a(n) ~ sqrt(145/sqrt(41)-21) * ((213+41*sqrt(41))/46)^n / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2014

A214693 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 4, 34, 338, 3691, 42623, 510949, 6289912, 78972928, 1006665781, 12985611054, 169115724583, 2219614920740, 29318819296959, 389331204757856, 5192978617937181, 69522908878900079, 933674035184058960, 12571898958515379108, 169651868248129552194
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 338*x^4 + 3691*x^5 + 42623*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^3-1)/A(x)^16 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^27 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^40 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^55 +...

Crossrefs

Cf. A214690, A214692, A214694, A214695, A181997 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9, which is the g.f. of row 3 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+6)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A214694 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(8*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 6, 69, 929, 13692, 213402, 3456450, 57585400, 980408857, 16982002433, 298322996205, 5302587890821, 95196447689434, 1723782813066284, 31447947375375315, 577509675356805547, 10667460556561578780, 198074286156460874227, 3695152948440645726312
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 929*x^4 + 13692*x^5 + 213402*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^9 + (A(x)-1)*(A(x)^3-1)/A(x)^20 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^33 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^48 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^65 +...

Crossrefs

Cf. A214690, A214692, A214693, A214695, A181998 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 -
1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(8*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 - 1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16, which is the g.f. of row 4 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+8)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A214695 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(10*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 8, 116, 1972, 36682, 722098, 14784834, 311629580, 6716892893, 147372681787, 3280609461927, 73912217824094, 1682234535898788, 38621258859241912, 893358073179541313, 20800314016777824187, 487100732909778007223, 11465386711990265812207
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

Compare the g.f. to the identity:
G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 116*x^3 + 1972*x^4 + 36682*x^5 + 722098*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^11 + (A(x)-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^39 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^56 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^75 +...

Crossrefs

Cf. A214690, A214692, A214693, A214694, A209441 (variant).

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 8*x^2 + 12*x^3 + 108*x^4 - 218*x^5 - 1938*x^6 - 834*x^7 + 27124*x^8 + 136919*x^9 + 393601*x^10 +
809873*x^11 + 1288950*x^12 + 1646268*x^13 + 1720788*x^14 + 1487263*x^15 + 1067345*x^16 + 635682*x^17 + 312646*x^18 + 125761*x^19 + 40734*x^20 +
10373*x^21 + 2001*x^22 + 275*x^23 + 24*x^24 + x^25 +x^2*O(x^n)), n))}
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(10*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y is a 25-degree polynomial in x and is the g.f. of row 5 in triangle A214690.
G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+10)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

A214670 Triangle, read by rows of n*(n+1)/2 terms, where row n equals the coefficients in the series reversion of the function G(x,n)-1 such that: x = Sum_{m>=1} 1/G(x,n)^(n*m) * Product_{k=1..m} (1 - 1/G(x,n)^k), for n>=1.

Original entry on oeis.org

1, 1, -1, -1, 1, -2, -1, 4, 4, 1, 1, -3, 0, 11, 1, -30, -42, -26, -8, -1, 1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1, 1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1
Offset: 1

Views

Author

Paul D. Hanna, Jul 25 2012

Keywords

Comments

The row sums are a signed version of A005014. [From _Olivier Gérard_, Jun 26 2012, in an email to the seqfan list, which suggested that the g.f. A(x,y) is a generalization of the g.f. for A005014.]

Examples

			Consider the family of power series G(x,n) that satisfy:
x = Sum_{m>=1} 1/G(x,n)^(n*m) * Product_{k=1..m} (1 - 1/G(x,n)^k).
Examples of sequences with g.f. G(x,n) are:
n=2: A001002 = [1, 1, 1, 3, 10, 38, 154, 654, 2871, 12925, ...];
n=3: A181997 = [1, 1, 2, 9, 46, 259, 1539, 9484, 59961, ...];
n=4: A181998 = [1, 1, 3, 18, 124, 935, 7443, 61510, 522467, ...];
n=5: A209441 = [1, 1, 4, 30, 260, 2463, 24656, 256493, 2745149, ...];
n=6: A209442 = [1, 1, 5, 45, 470, 5365, 64766, 813012, 10505163, ...]; ...
Observe that Series_Reversion( G(x,n) - 1 ) is given by the polynomials:
n=1: x;
n=2: x - x^2 - x^3;
n=3: x - 2*x^2 - x^3 + 4*x^4 + 4*x^5 + x^6;
n=4: x - 3*x^2 + 11*x^4 + x^5 - 30*x^6 - 42*x^7 - 26*x^8 - 8*x^9 - x^10;
n=5: x - 4*x^2 + 2*x^3 + 20*x^4 - 19*x^5 - 100*x^6 + 3*x^7 + 403*x^8 + 808*x^9 + 861*x^10 + 584*x^11 + 262*x^12 + 76*x^13 + 13*x^14 + x^15; ...
This triangle of coefficients in the above polynomials begins:
[1];
[1, -1, -1];
[1, -2, -1, 4, 4, 1];
[1, -3, 0, 11, 1, -30, -42, -26, -8, -1];
[1, -4, 2, 20, -19, -100, 3, 403, 808, 861, 584, 262, 76, 13, 1];
[1, -5, 5, 30, -65, -191, 378, 1557, 103, -8551, -23911, -37958, -41831, -34156, -21179, -10015, -3571, -933, -169, -19, -1];
[1, -6, 9, 40, -145, -261, 1384, 2897, -8980, -38710, -14146, 258401, 990407, 2170834, 3426095, 4198850, 4137440, 3336534, 2220430, 1221799, 554027, 205250, 61206, 14351, 2550, 323, 26, 1];
[1, -7, 14, 49, -266, -245, 3325, 2596, -36710, -70556, 281645, 1413916, 1184890, -10255248, -54012830, -156371880, -329973512, -552895722, -765517470, -895408431, -896614676, -774834055, -580511469, -377792286, -213512611, -104550572, -44163315, -15985147, -4910774, -1263620, -267378, -45321, -5918, -559, -34, -1]; ...

Links

Crossrefs

Cf. A001002, A181997, A181998, A209441, A209442, A005014 (row sums), A214669.
Cf. A214690 (variant).

Programs

  • PARI
    {T(n,k)=local(Axy=x*y);Axy=sum(m=1,n,-x^m*prod(j=1,m,(1-(1+y)^j)/(1-x*(1+y)^j)+x*O(x^n)));polcoeff(polcoeff(Axy,n,x),k,y)}
{for(n=1,10,for(k=1,n*(n+1)/2,print1(T(n,k),", "));print(""))}
  • PARI
    {a(n,p)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(p*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
{for(n=1,8,Tn=Vec(serreverse(sum(m=1,n*(n+1)/2,a(m,n)*x^m)+x*O(x^(n*(n+1)/2))));for(k=1,n*(n+1)/2,print1(Tn[k],", "));print(""))}

Formula

G.f.: A(x,y) = Sum_{n>=1} -x^n * Product_{k=1..n} (1 - (1+y)^k) / (1 - x*(1+y)^k).
G.f. for row n is R(y,n) = Sum_{k=1..n*(n+1)/2} y^k*T(n,k) defined by:
A(x,y) = Sum_{n>=1} x^n * R(y,n) such that:
R(y,n) = Series_Reversion( G(y,n) - 1 ) where G(y,n) satisfies:
y = Sum_{m>=1} 1/G(y,n)^(n*m) * Product_{k=1..m} (1 - 1/G(y,n)^k), for n>=1.
Row polynomials R(y,n) satisfy:
(1) R(1,n) = (-1)^(n-1) * A005014(n) for n>=1.
(2) R(-1,n) = 1 for n>=1.
(3) R'(-1,n) = 0 for n>1.
(4) R'(1,n) = A214669(n) for n>=1.

A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, 3, 21, 172, 1557, 14937, 148870, 1523150, 15874211, 167584946, 1784250269, 19082848084, 204183773733, 2174724531143, 22887441573480, 235016048710027, 2294441979279215, 19936497820248076, 118333942636382173, -709004900481995789, -49850788347995316262
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247482 (exponent=0), A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF^5,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

Formula

a(n) ~ c * 12^n * n^(n-2) / (exp(n) * Pi^(2*n)), where c = -sqrt(6) * Pi^3 * exp(5*Pi^2/24)/24 = -24.7341070998048267... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A247481 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^n * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, -1, -1, -2, -14, -98, -822, -7948, -86590, -1046916, -13892842, -200653570, -3133064534, -52596852266, -944892417438, -18091297436248, -367841660947508, -7916992964642992, -179849204152350892, -4300928485463624458, -108013481381638292266
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247482 (exponent=0), A249934 (exponent=3), A214692 (exponent=4), A247480 (exponent=5), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

Formula

a(n) ~ c * 12^n * n^n / (exp(n) * Pi^(2*n)), where c = -2*sqrt(6)/(Pi*exp(Pi^2/8)) = -0.45411558500969644... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A247482 G.f. A(x) satisfies: x = Sum_{n>=1} Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, -2, 1, -3, -18, -124, -1174, -12150, -141536, -1816780, -25461723, -386593670, -6320496592, -110711177281, -2068814967831, -41089562943757, -864563028340432, -19214971769126974, -449887669808788433, -11069673481210168218, -285604488897863640237
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 01 2014

Keywords

Links

Crossrefs

Cf. A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A247480 (exponent=5), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).
Cf. A214690, A214670.

Programs

  • Mathematica
    nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1)),{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]
  • PARI
    {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=-polcoeff(sum(m=1, #A, prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2024, after Paul D. Hanna

Formula

a(n) ~ c * 12^n * n^(n+1/2) / (exp(n) * Pi^(2*n)), where c = -12 / (Pi^(3/2) * exp(5*Pi^2/24)) = -0.275723765924812729... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A214691 G.f.: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} (2^(2*k-1) - 1) / (1 + 2^(2*k-1)*x).

Original entry on oeis.org

1, 1, 5, 151, 19025, 9702751, 19851828545, 162586475783551, 5327308465523832065, 698250320576208668759551, 366082867573618138109269955585, 767730685732013278335855487355082751, 6440190236715680978727827356359771295535105
Offset: 0

Views

Author

Paul D. Hanna, Jul 26 2012

Keywords

Comments

A variant of A005014. Equals row sums (unsigned) of triangle A214690.

Crossrefs

Cf. A005014 (variant), A214690.

Programs

  • PARI
    {a(n)=if(n==0,1,2*(4^(n-1)-1)*a(n-1) - (-1)^n)}
  • PARI
    {a(n)=local(A=x); A=sum(m=0, n, x^m*prod(j=1, m, (2^(2*j-1)-1)/(1+x*2^(2*j-1))+x*O(x^n))); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "));

Formula

a(n) = 2*(4^(n-1) - 1)*a(n-1) - (-1)^n for n>0 with a(0)=1.
Showing 1-9 of 9 results.

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