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 1-3 of 3 results.

A211855 G.f. satisfies: A(x) = (1+x*A(x)^3)*(1+x^2*A(x)^2)*(1+x^3*A(x)).

Original entry on oeis.org

1, 1, 4, 19, 98, 553, 3288, 20287, 128681, 833889, 5496837, 36742204, 248454438, 1696588460, 11682677436, 81031854579, 565614332353, 3970182041035, 28006229772030, 198438070511163, 1411652452459443, 10078529348799106, 72192155099054325, 518659038159324250
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 19*x^3 + 98*x^4 + 553*x^5 + 3288*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 46*x^3 + 250*x^4 + 1454*x^5 + 8827*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 82*x^3 + 468*x^4 + 2808*x^5 + 17431*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 128*x^3 + 765*x^4 + 4736*x^5 + 30086*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 185*x^3 + 1155*x^4 + 7376*x^5 + 47970*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 254*x^3 + 1653*x^4 + 10884*x^5 + 72474*x^6 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^2 + x^3*(A(x)+A(x)^5) + x^4*A(x)^4 + x^5*A(x)^3 + x^6*A(x)^6.

Links

Crossrefs

Cf. A182053, A211854.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A^3)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

a(n) ~ sqrt(s*(2*r*s + s^2 + 5*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + 3*r^2*(1 + s^4)) / (Pi*(r + 3*s + 3*r^4*s + 6*r^3*s^2 + 10*r^2*s^3 + 15*r^5*s^4))) / (2*n^(3/2)*r^n), where r = 0.1303652752058746790368151406944165350206179676971... and s = 1.504659035764367744283558911063644754705733371817... are real roots of the system of equations (1 + r^3*s)*(1 + r^2*s^2)*(1 + r*s^3) = s, r*(2*r*s + 3*s^2 + 3*r^4*s^2 + 4*r^3*s^3 + 6*r^5*s^5 + r^2*(1 + 5*s^4)) = 1. - Vaclav Kotesovec, Nov 22 2017

A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).

Original entry on oeis.org

1, 1, 2, 6, 16, 46, 140, 435, 1382, 4474, 14687, 48787, 163703, 554009, 1888794, 6481220, 22366415, 77575617, 270277602, 945480612, 3319582632, 11693824752, 41318554495, 146399071577, 520042511448, 1851657641932, 6607352892709, 23624965371264
Offset: 0

Views

Author

Paul D. Hanna, Apr 22 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 46*x^5 + 140*x^6 + 435*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 148*x^5 + 472*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 102*x^4 + 336*x^5 + 1124*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 185*x^4 + 648*x^5 + 2272*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*(A(x) + A(x)^3) + x^4*A(x)^2 + x^5*A(x)^3 + x^6*A(x)^4.

Links

Crossrefs

Cf. A182053, A211854, A211855.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

a(n) ~ sqrt(s*(1 + 2*r*s + 4*r^3*s + 5*r^4*s^2 + 6*r^5*s^3 + 3*r^2*(1 + s^2)) / (Pi*(1 + r^2 + 3*r*s + 3*r^3*s + 6*r^4*s^2))) / (2 * n^(3/2) * r^(n + 1/2)), where r = 0.2649675733882333627400730579639429790476557486165... and s = 2.383929237709193665917448862090331200952809331679... are roots of the system of equations (1 + r*s)*(1 + r^3*s)*(1 + r^2*s^2) = s, r*(1 + r^2 + 2*r*s + 2*r^3*s + 3*r^2*s^2 + 3*r^4*s^2 + 4*r^5*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017

A212070 G.f. satisfies: A(x) = (1+x*A(x))*(1+x*A(x)^2)*(1+x*A(x)^3).

Original entry on oeis.org

1, 3, 21, 199, 2166, 25551, 317736, 4101292, 54429850, 738053745, 10180705447, 142408547576, 2015296793331, 28800644332829, 415060115307920, 6025247760182629, 88023011490624217, 1293147320502884759, 19092299095314415811, 283137984006724444796
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 199*x^3 + 2166*x^4 + 25551*x^5 +..
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 524*x^3 + 5967*x^4 + 72456*x^5 +...
A(x)^3 = 1 + 9*x + 90*x^2 + 1002*x^3 + 11970*x^4 + 150057*x^5 +...
A(x)^4 = 1 + 12*x + 138*x^2 + 1660*x^3 + 20823*x^4 + 269964*x^5 +...
A(x)^5 = 1 + 15*x + 195*x^2 + 2525*x^3 + 33255*x^4 + 446298*x^5 +...
A(x)^6 = 1 + 18*x + 261*x^2 + 3624*x^3 + 50076*x^4 + 695934*x^5 +...
where A(x) = 1 + x*A(x) + x*A(x)^2 + x*(1+x)*A(x)^3 + x^2*A(x)^4 + x^2*A(x)^5 + x^3*A(x)^6.

Links

Crossrefs

Cf. A182053, A211854.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x*A^2)*(1+x*A^3)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: A(x) = 1 + x*A(x)*(1+A(x)+A(x)^2) + x^2*A(x)^3*(1+A(x)+A(x)^2) + x^3*A(x)^6.
a(n) ~ sqrt((s*(1 + s + (1 + 2*r)*s^2 + 2*r*s^3 + 2*r*s^4 + 3*r^2*s^5)) / (1 + 3*(1 + r)*s + 6*r*s^2 + 10*r*s^3 + 15*r^2*s^4)) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.06228198686712455165459532624572875420874352588006064829276... and s = 1.61944833450852965640457413211207525783408084239130679443147... are roots of the system of equations (1 + r*s) * (1 + r*s^2) * (1 + r*s^3) = s, r*(1 + 2*s + 3*(1+r)*s^2 + 4*r*s^3 + 5*r*s^4 + 6*r^2*s^5) = 1. - Vaclav Kotesovec, Aug 24 2017
Showing 1-3 of 3 results.

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