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 1-8 of 8 results.

A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2.

Original entry on oeis.org

1, 1, 4, 8, 28, 80, 308, 984, 3980, 13472, 56164, 197032, 838396, 3013872, 13015188, 47624568, 207971436, 771336512, 3397886660, 12736715592, 56502898140, 213618833808, 953139545076, 3629043226392, 16270547827020, 62317467147744
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Comments

Specific values: A(2/9) = 17/9 and A(-2/9) = 17/18.
Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...
with A(r) = (2 + sqrt(1-3*r))*(1+r^2)/(1+r) = 2.19775350...
and A(-r) = (2 - sqrt(1+3*r))*(1+r^2)/(1-r) = 3*(1+r^2) - A(r) = 0.9569470...
At x=r, the equation (*) (1+x^2)^2 - 2*(1+x^2)*y + (1+x)*y^2 - x*y^3 = 0, which is satisfied by y = A(x), factors out to: (y - A(r))^2 * (y - A(r)*(1+r^2)/(2*(A(r)-1-r^2))) = 0; this gives the relation: (A(r)-1-r^2)*(3+3*r^2-A(r)) = r*A(r)^2. At x>r, the equation (*) admits complex solutions for y.

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +...
where 1 - (1+x^2)/A(x) = x*A(x)/A(-x).
Related expansions:
A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...
where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x).

Crossrefs

Cf. A143339, A143554, A143556, A143557, A143558, A143559.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^2/subst(A^2,x,-x));polcoeff(A,n)}

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0.
(2) A(x) = 1 + x*A(x)^2 + x^2 + x^2*A(-x).
(3) A(x) = 1 + x^2 + x*A(x)^2/A(-x).
(4) A(x) = 1 + x^2/(1 - A(-x)).
(5) A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x.
(6) A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^2/G(-x) is the g.f. of A143339.
Recurrence: (n-1)*(n+1)*(4*n^3 - 32*n^2 + 71*n - 30)*a(n) = 6*(8*n^3 - 56*n^2 + 101*n - 10)*a(n-1) + 6*(12*n^5 - 132*n^4 + 499*n^3 - 700*n^2 + 102*n + 305)*a(n-2) - 18*(n-4)*(8*n - 25)*a(n-3) + 27*(n-5)*(n-4)*(4*n^3 - 20*n^2 + 19*n + 13)*a(n-4). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ c * 3^(n-1) * 2*sqrt(6*sqrt(3)-6 + sqrt(9+6*sqrt(3))) / (2*sqrt(Pi) * (2*sqrt(3)-3)^(n/2+1/4) * n^(3/2)), where c = 4/(2+12^(1/4)) if n is even and c = 12/(6+12^(3/4)) if n is odd. - Vaclav Kotesovec, Dec 29 2013

A212527 G.f. satisfies: A(x) = 1 + x*A(x)^2 / (A(I*x) * A(-I*x)).

Original entry on oeis.org

1, 1, 2, 8, 26, 56, 194, 832, 2866, 7904, 30690, 137000, 497706, 1491512, 6041602, 27557184, 102985186, 321675648, 1333006018, 6160815624, 23426000186, 75016874488, 315357132994, 1470462300160, 5656904907026, 18419315779552, 78201118018466, 366962271138472
Offset: 0

Views

Author

Paul D. Hanna, Jun 13 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 56*x^5 + 194*x^6 + 832*x^7 +...
Related expansions begin:
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 196*x^5 + 668*x^6 + 2692*x^7 +...
A(I*x)*A(-I*x) = 1 - 3*x^2 + 40*x^4 - 316*x^6 + 4624*x^8 - 50676*x^10 + 811192*x^12 -+...

Crossrefs

Cf. A143339, A216681, A216712, A216713.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^2/(subst(A,x,I*x+x*O(x^n))*subst(A,x,-I*x+x*O(x^n))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A216713 G.f.: A(x) = 1 + x*A(x)^2 / ( A(w*x)*A(w^2*x) ), where w = exp(2*Pi*I/3).

Original entry on oeis.org

1, 1, 3, 12, 27, 105, 420, 1242, 5295, 22395, 72738, 323268, 1410684, 4806675, 21881721, 97371786, 341608239, 1579726122, 7123796790, 25489388367, 119184247992, 542664427242, 1969440159591, 9284827569117, 42584603672868, 156213604844883, 741154831030785
Offset: 0

Views

Author

Paul D. Hanna, Sep 14 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 27*x^4 + 105*x^5 + 420*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 87*x^4 + 336*x^5 + 1356*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 189*x^4 + 756*x^5 + 3132*x^6 +...
Let w = exp(2*Pi*I/3), then A(x) = 1 + x*A(x)^3/(A(x)*A(w*x)*A(w^2*x)) where
A(x)*A(w*x)*A(w^2*x) = 1 + 28*x^3 + 1134*x^6 + 61857*x^9 + 3929121*x^12 + 272388420*x^15 + 19981576476*x^18 + 1524888581787*x^21 +...

Links

Crossrefs

Cf. A143339, A212527, A216712, A216681.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n+1,A=1+x*A^3*exp(-3*sum(m=1,n\3,x^(3*m)*polcoeff(log(A),3*m))+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A143562 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^2.

Original entry on oeis.org

1, 1, 5, 17, 105, 481, 3261, 16801, 119697, 656129, 4819061, 27447601, 205776121, 1202943457, 9152680109, 54524185409, 419491297313, 2534963932417, 19673179986661, 120224135048273, 939543098579081, 5793676726569697
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 5*x^2 + 17*x^3 + 105*x^4 + 481*x^5 + 3261*x^6 +...
A(x)*A(-x) = 1 + 9*x^2 + 201*x^4 + 6321*x^6 + 233073*x^8 +...

Crossrefs

Cf. A143556, A143339.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^3/subst(A^2,x,-x));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) + A(-x) = 1 + (1+x^2)*A(x)*A(-x).
G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(2*n) / A(-x)^(2*n).
G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x)^(2*n)/A(-x)^(2*n) * x^n/n ).
G.f.: A(x) = G(x)/(1+x^2) where G(x) = 1 + x*G(x)^3/G(-x)^3 is the g.f. of A143556.

A216712 G.f.: A(x) = 1 + x*A(x)^3 / ( A(-x)*A(I*x)*A(-I*x) ), where I^2 = -1.

Original entry on oeis.org

1, 1, 4, 22, 140, 514, 3444, 23790, 165932, 774610, 5767268, 42526198, 310791884, 1574532626, 12230311188, 92980917006, 696528653740, 3677761305954, 29231321098692, 226211978983190, 1720430261953036, 9313977313216354, 75106192841523892, 588010633850768622
Offset: 0

Views

Author

Paul D. Hanna, Sep 14 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 514*x^5 + 3444*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 1484*x^5 + 9520*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 3024*x^5 + 19240*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 5264*x^5 + 33800*x^6 +...
A(x)*A(-x) = 1 + 7*x^2 + 252*x^4 + 6496*x^6 + 308820*x^8 + 10966136*x^10 + 582452652*x^12 + 23322250960*x^14 + 1309365750212*x^16 +...
Note that A(x) = 1 + x*A(x)^4/(A(x)*A(-x)*A(I*x)*A(-I*x)) where
A(x)*A(-x)*A(I*x)*A(-I*x) = 1 + 455*x^4 + 590200*x^8 + 1124826664*x^12 + 2538673877080*x^16 + 6294363022919816*x^20 + 16568529053651321656*x^24 +...
Note also that a bisection of 1/A(x)^3 equals a bisection of 1/A(x)^4:
1/A(x)^3 = 1 - 3*x - 6*x^2 - 28*x^3 - 165*x^4 + 273*x^5 - 2292*x^6 +...
1/A(x)^4 = 1 - 4*x - 6*x^2 - 28*x^3 - 165*x^4 + 728*x^5 - 2292*x^6 +...

Crossrefs

Cf. A143339, A216681, A212527, A216713.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n,A=1+x*A^3/(subst(A,x,-x)*subst(A,x,I*x)*subst(A,x,-I*x)));polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=1,n+1,A=1+x*A^4*exp(-4*sum(m=1,n\4,x^(4*m)*polcoeff(log(A),4*m))+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A143340 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x).

Original entry on oeis.org

1, 1, 4, 15, 84, 402, 2520, 13339, 88484, 494814, 3395816, 19657398, 137999048, 818024484, 5836517808, 35201610387, 254231733188, 1553691459558, 11327637588552, 69948932919906, 513856752260184, 3199802098978428
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2008

Keywords

Examples

			A bisection of g.f. A(x) equals a bisection of A(x)^3:
A(x) = 1 + x + 4*x^2 + 15*x^3 + 84*x^4 + 402*x^5 + 2520*x^6 + 13339*x^7 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 70*x^3 + 402*x^4 + 2163*x^5 + 13339*x^6 +...
so that A(x) - x*A(x)^3 = 1 + x^2*[A(x)*A(-x)]^2, where
[A(x)*A(-x)]^2 = 1 + 14*x^2 + 357*x^4 + 11522*x^6 + 420170*x^8 +...
A(x)*A(-x) = 1 + 7*x^2 + 154*x^4 + 4683*x^6 + 165446*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^2/A(-x) + x^2*A(x)^4/A(-x)^2 + x^3*A(x)^6/A(-x)^3 +...
log(A(x)) = x*A(x)^2/A(-x) + x^2/2*A(x)^4/A(-x)^2*x^2 + x^3/3*A(x)^6/A(-x)^3 +...

Crossrefs

Cf. A143339, A143341.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A^3/subst(A,x,-x));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*A^(2*m)/subst(A^m,x,-x+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,A^(2*m)*subst(A^-m,x,-x)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) - x*A(x)^3 = 1 + x^2*[A(x)*A(-x)]^2.
G.f. satisfies:
A(x) = Sum{n>=1} x^n * A(x)^(2*n)/A(-x)^n ;
A(x) = exp( Sum{n>=1} x^n/n * A(x)^(2*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]

A143341 G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x).

Original entry on oeis.org

1, 1, 5, 26, 195, 1303, 11076, 81910, 740151, 5782175, 54176573, 438029432, 4203769940, 34798104500, 339699218160, 2860590892318, 28283147265023, 241296800029199, 2409437282086511, 20767852798378330, 209017295575667771
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2008

Keywords

Examples

			A bisection of g.f. A(x) equals a bisection of A(x)^4:
A(x) = 1 + x + 5*x^2 + 26*x^3 + 195*x^4 + 1303*x^5 + 11076*x^6 + 81910*x^7 +...
A(x)^4 = 1 + 4*x + 26*x^2 + 168*x^3 + 1303*x^4 + 9744*x^5 + 81910*x^6 +...
so that A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3, where
[A(x)*A(-x)]^3 = 1 + 27*x^2 + 1332*x^4 + 82791*x^6 + 5800329*x^8 +...
A(x)*A(-x) = 1 + 9*x^2 + 363*x^4 + 20820*x^6 + 1397511*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^3/A(-x) + x^2*A(x)^6/A(-x)^2 + x^3*A(x)^9/A(-x)^3 +...
log(A(x)) = x*A(x)^3/A(-x) + x^2/2*A(x)^6/A(-x)^2*x^2 + x^3/3*A(x)^9/A(-x)^3 +...

Crossrefs

Cf. A143339, A143340.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A^4/subst(A,x,-x));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*A^(3*m)/subst(A^m,x,-x+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,A^(3*m)*subst(A^-m,x,-x)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) - x*A(x)^4 = 1 + x^2*[A(x)*A(-x)]^3.
G.f. satisfies:
A(x) = Sum{n>=1} x^n * A(x)^(3*n)/A(-x)^n;
A(x) = exp( Sum{n>=1} x^n/n * A(x)^(3*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]

A143342 G.f. satisfies: A(x) = 1 + x*A(x)^5/A(-x).

Original entry on oeis.org

1, 1, 6, 40, 374, 3215, 34298, 326360, 3710278, 37289620, 440121880, 4577214736, 55375589594, 589530372890, 7258264793564, 78597770766160, 980423896907046, 10754940952651740, 135521929778850952, 1501817992511869280
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2008

Keywords

Comments

More generally, if A(x) = 1 + x*A(x)^n/A(-x)
then A(x) - x*A(x)^n = 1 + x^2*[A(x)*A(-x)]^(n-1)
so that a bisection of A(x) equals a bisection of A(x)^n.

Examples

			A bisection of g.f. A(x) equals a bisection of A(x)^5:
A(x) = 1 + x + 6*x^2 + 40*x^3 + 374*x^4 + 3215*x^5 + 34298*x^6 + 326360*x^7 +...
A(x)^5 = 1 + 5*x + 40*x^2 + 330*x^3 + 3215*x^4 + 30756*x^5 + 326360*x^6 +...
so that A(x) - x*A(x)^5 = 1 + x^2*[A(x)*A(-x)]^4, where
[A(x)*A(-x)]^4 = 1 + 44*x^2 + 3542*x^4 + 358468*x^6 + 40846025*x^8 + +...
A(x)*A(-x) = 1 + 11*x^2 + 704*x^4 + 65054*x^6 + 7062088*x^8 +...
Related expressions.
A(x) = 1 + x*A(x)^4/A(-x) + x^2*A(x)^8/A(-x)^2 + x^3*A(x)^12/A(-x)^3 +...
log(A(x)) = x*A(x)^4/A(-x) + x^2/2*A(x)^8/A(-x)^2*x^2 + x^3/3*A(x)^12/A(-x)^3 +...

Crossrefs

Cf. A143339, A143340, A143341.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A^5/subst(A,x,-x));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*A^(4*m)/subst(A^m,x,-x+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(sum(m=1,n,A^(4*m)*subst(A^-m,x,-x)*x^m/m)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) - x*A(x)^5 = 1 + x^2*[A(x)*A(-x)]^4.
G.f. satisfies:
A(x) = Sum{n>=1} x^n * A(x)^(4*n)/A(-x)^n;
A(x) = exp( Sum{n>=1} x^n/n * A(x)^(4*n)/A(-x)^n ). [From Paul D. Hanna, Sep 30 2011]
Showing 1-8 of 8 results.

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