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 1-5 of 5 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

Views

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

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.

A143559 G.f. satisfies: A(x) = 1 + x*A(x)^6/A(-x)^6.

Original entry on oeis.org

1, 1, 12, 72, 1012, 9552, 148764, 1609496, 26398020, 305821344, 5174354988, 62479377384, 1079265357204, 13399747245040, 234917433809724, 2975608178304696, 52748683164797668, 678307369324850496
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 12*x^2 + 72*x^3 + 1012*x^4 + 9552*x^5 + 148764*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 122*x^3 + 242*x^4 + 16002*x^5 + 38962*x^6 +...
A(x)^5/A(-x)^5 = 1 + 10*x + 50*x^2 + 770*x^3 + 6450*x^4 + 109802*x^5 +...
where 1 - (1+x^2)/A(x) = x*A(x)^5/A(-x)^5.

Crossrefs

Cf. A143555, A143556, A143557, A143558.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^6/subst(A^6,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)^6/A(-x)^5.
G.f. satisfies: (A(x) - 1)^5 = ( 1 - (1+x^2)/A(x) )^6/x = x^5*A(x)^30/A(-x)^30.
G.f.: A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^6/G(-x)^5.

A143561 G.f. satisfies: A(x) = ( 1 + x*A(x)/A(-x) )^2.

Original entry on oeis.org

1, 2, 9, 24, 88, 280, 1064, 3672, 14456, 52184, 210504, 782232, 3210904, 12176792, 50638440, 194956248, 818961080, 3189915224, 13508052104, 53105011480, 226355549400, 896636646936, 3842662060200, 15317408281944, 65946510374136
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...
A(x)/A(-x) = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 + 984*x^6 +...

Crossrefs

Cf. A143555.

Programs

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

Formula

G.f. satisfies: (1+x^2)^2 - 2*(1+x^2)*G(x) + (1+x)*G(x)^2 - x*G(x)^3 = 0 where G(x)^2 = A(x) and G(x) = 1 + x*A(x)/A(-x) is the g.f. of A143555.
Showing 1-5 of 5 results.

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