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 1-4 of 4 results.

A143556 G.f. satisfies: A(x) = 1 + x*A(x)^3/A(-x)^3.

Original entry on oeis.org

1, 1, 6, 18, 110, 498, 3366, 17282, 122958, 672930, 4938758, 28103730, 210595182, 1230391058, 9358456230, 55727128866, 428643977422, 2589488117826, 20092671283974, 122759098980690, 959216278565742, 5913900861617970
Offset: 0

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Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 6*x^2 + 18*x^3 + 110*x^4 + 498*x^5 + 3366*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 26*x^3 + 50*x^4 + 706*x^5 + 1650*x^6 +...
A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 60*x^3 + 208*x^4 + 1716*x^5 +...
where 1 - (1+x^2)/A(x) = x*A(x)^2/A(-x)^2.

Links

Crossrefs

Cf. A143562, A143555, A143557, A143558, A143559.

Programs

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

Formula

G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)).
G.f. satisfies: A(x) = 1 + x^2 + x*A(x)^3/A(-x)^2.
G.f. satisfies: (A(x) - 1)^2 = ( 1 - (1+x^2)/A(x) )^3/x = x^2*A(x)^6/A(-x)^6.
G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^3/G(-x)^2 is the g.f. of A143562.
G.f. satisfies: x*A(x)^5 - 2*x*A(x)^4 - (1-x)*A(x)^3 + 3*(1+x^2)*A(x)^2 - 3*(1+x^2)^2*A(x) + (1+x^2)^3 = 0.
Recurrence: 4*(n-1)*n*(2*n-5)*(2*n+1)*(2916*n^10 - 99630*n^9 + 1494855*n^8 - 12945798*n^7 + 71493183*n^6 - 262308129*n^5 + 645244282*n^4 - 1046448887*n^3 + 1066283852*n^2 - 614660500*n + 152638416)*a(n) = 60*(n-1)*(13122*n^11 - 458217*n^10 + 7044759*n^9 - 62741439*n^8 + 358008636*n^7 - 1365100815*n^6 + 3513825159*n^5 - 6010387373*n^4 + 6521940316*n^3 - 4078695988*n^2 + 1207261712*n - 113170176)*a(n-1) + 15*(n-2)*(160380*n^13 - 6121170*n^12 + 104460435*n^11 - 1051745310*n^10 + 6938544798*n^9 - 31476010053*n^8 + 100128993299*n^7 - 223244300184*n^6 + 341877397736*n^5 - 343306364591*n^4 + 206330136024*n^3 - 62025904772*n^2 + 8101283136*n - 2665897920)*a(n-2) + 450*(n-4)*(7020*n^10 - 107820*n^9 + 91377*n^8 + 9009842*n^7 - 87380558*n^6 + 404731832*n^5 - 1079876519*n^4 + 1690685386*n^3 - 1439622136*n^2 + 509372600*n + 4226320)*a(n-3) + 750*(n-5)*(n-4)*(14580*n^12 - 498150*n^11 + 7512345*n^10 - 65844630*n^9 + 371440818*n^8 - 1409248026*n^7 + 3643384398*n^6 - 6348642805*n^5 + 7178246227*n^4 - 4869145209*n^3 + 1716210104*n^2 - 292182404*n + 75613440)*a(n-4) + 3750*(n-6)*(n-5)*(n-4)*(3240*n^8 - 52785*n^7 + 324459*n^6 - 854916*n^5 + 387102*n^4 + 2695803*n^3 - 5239793*n^2 + 2713946*n + 268800)*a(n-5) + 3125*(n-7)*(n-6)*(n-5)*(n-4)*(2916*n^10 - 70470*n^9 + 729405*n^8 - 4223718*n^7 + 14971977*n^6 - 33317457*n^5 + 45697282*n^4 - 36099439*n^3 + 14258060*n^2 - 2573132*n + 694560)*a(n-6). - Vaclav Kotesovec, Mar 25 2014
a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where {r1 = r = 0.13384151194121538538097804723..., s1 = 1.57588974374012701113388456...} and {r2 = -r, s2 = 0.9688941320566492403600185...} are roots of the system of equations r*(r^5 + 3*r*(s-1)^2 + (s-1)^2*s^3) = 3*r^4*(s-1) + (s-1)^3, r*(s-1)*(6*r + s^2*(5*s-3)) = 3*(r^4 + (s-1)^2), and c = c1+c2 = 0.525673619703566161096484... if n is even, and c = c1-c2 = 0.471796676012154625609556... if n is odd, where c1 = M(r1,s1), c2=M(r2,s2), and M(r,s) = sqrt(r*(6*r^5 - 12*r^3*(s-1) + 6*r*(s-1)^2 + (s-1)^2*s^3)/(3+3*r^2-3*s+r*s*(3-12*s+10*s^2)))/2. - Vaclav Kotesovec, Mar 25 2014

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

A143557 G.f. satisfies: A(x) = 1 + x*A(x)^4/A(-x)^4.

Original entry on oeis.org

1, 1, 8, 32, 280, 1728, 16744, 117856, 1202552, 9044352, 95203784, 745451168, 8011827928, 64459117632, 703166465320, 5769038826208, 63639465830712, 529889242505984, 5896324892061576, 49665617425122592, 556508207889107096
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 8*x^2 + 32*x^3 + 280*x^4 + 1728*x^5 + 16744*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 50*x^3 + 98*x^4 + 2658*x^5 + 6370*x^6 +...
A(x)^3/A(-x)^3 = 1 + 6*x + 18*x^2 + 182*x^3 + 930*x^4 + 10374*x^5 +...
where 1 - (1+x^2)/A(x) = x*A(x)^3/A(-x)^3.

Crossrefs

Cf. A143564, A143555, A143556, A143558, A143559.

Programs

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

Formula

G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)).
G.f. satisfies: A(x) = 1 + x^2 + x*A(x)^4 / A(-x)^3.
G.f. satisfies: (A(x) - 1)^3 = ( 1 - (1+x^2)/A(x) )^4/x = x^3*A(x)^12/A(-x)^12.
G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^4/G(-x)^3 is the g.f. of A143564.

A143558 G.f. satisfies: A(x) = 1 + x*A(x)^5/A(-x)^5.

Original entry on oeis.org

1, 1, 10, 50, 570, 4450, 56202, 501970, 6676410, 63799490, 875391370, 8715058802, 122088479930, 1249437863970, 17764858122250, 185445650940690, 2666213981716282, 28252030821781890, 409717783914784010
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 10*x^2 + 50*x^3 + 570*x^4 + 4450*x^5 + 56202*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 82*x^3 + 162*x^4 + 7202*x^5 + 17442*x^6 +...
A(x)^4/A(-x)^4 = 1 + 8*x + 32*x^2 + 408*x^3 + 2752*x^4 + 38760*x^5 +...
where 1 - (1+x^2)/A(x) = x*A(x)^4/A(-x)^4.

Crossrefs

Cf. A143555, A143556, A143557, A143559.

Programs

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

Formula

G.f. satisfies: A(x) = 1 + x^2/(1 - A(-x)).
G.f. satisfies: A(x) = 1 + x^2 + x*A(x)^5/A(-x)^4.
G.f. satisfies: (A(x) - 1)^4 = ( 1 - (1+x^2)/A(x) )^5/x = x^4*A(x)^20/A(-x)^20.
G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^5/G(-x)^4.
Showing 1-4 of 4 results.

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