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.

A276310 G.f. A(x) satisfies: x = A(x)-2*A(x)^2-2*A(x)^3.

Original entry on oeis.org

1, 2, 10, 60, 404, 2912, 21984, 171600, 1373680, 11215776, 93039648, 781936896, 6643741440, 56973685760, 492482782208, 4286561051904, 37536888622848, 330471001126400, 2923338431270400, 25970490200202240, 231607762146309120, 2072719382680535040
Offset: 1

Views

Author

Tom Richardson, Aug 29 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 404*x^5 + 2912*x^6 + 21984*x^7 +...
Related expansions.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1148*x^6 + 8640*x^7 + 67296*x^8 +...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2352*x^7 + 18504*x^8 +...
where x = A(x) - 2*A(x)^2 - 2*A(x)^3.

Links

Crossrefs

Cf. A250886, A276314, A276315, A276316.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - 2*x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)
  • PARI
    {a(n)=polcoeff(serreverse(x - 2*x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f.: Series_Reversion(x - 2*x^2 - 2*x^3).
Conjecture: 3*n*(n-1)*a(n) -13*(n-1)*(2*n-3)*a(n-1) -3*(3*n-5)*(3*n-7)*a(n-2)=0. - R. J. Mathar, Sep 17 2016
a(n) ~ (13 + 5*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A276314 G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

Original entry on oeis.org

1, 1, 5, 20, 104, 546, 3066, 17655, 104555, 630773, 3867617, 24020932, 150827740, 955808680, 6105327912, 39268000188, 254093573088, 1652984379150, 10804631902350, 70925539707330, 467373389649870, 3090558380977020, 20501504119375500, 136392970090612950
Offset: 1

Views

Author

Tom Richardson, Aug 29 2016

Keywords

Examples

			G.f.: A(x) = x+x^2+5*x^3+20*x^4+104*x^5+546*x^6+3066*x^7+... Related Expansions:
A(x)^2=x^2+2*x^3+11*x^4+50*x^5+273*x^6+1500*x^7+8664*x^8+...
A(x)^3=x^3+3*x^4+18*x^5+91*x^6+522*x^7+2997*x^8+17831*x^9+...

Links

Crossrefs

Cf. A250886, A276310, A276315, A276316.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - x^2 - 3*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)
  • PARI
    {a(n)=polcoeff(serreverse(x - x^2 - 3*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f.: Series_Reversion(x-x^2-3*x^3)
Conjecture: +169*n*(n+2)*(n-1)*a(n) +13*(n-1) *(13*n^2+26*n-220) *a(n-1) +(-7277*n^3+13423*n^2+43814*n-81700) *a(n-2) -27*(3*n-10) *(3*n-8) *(71*n+197)*a(n-3)=0. - R. J. Mathar, Sep 17 2016
a(n) ~ (29 + 20*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 13^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017

A276315 G.f. A(x) satisfies: x = A(x)-3*A(x)^2-2*A(x)^3.

Original entry on oeis.org

1, 3, 20, 165, 1524, 15078, 156264, 1674585, 18404980, 206325834, 2350049208, 27118926354, 316381296840, 3725407768140, 44217602683728, 528470024711841, 6354463541900148, 76818345766932450, 933089010748085400, 11382500895815005110, 139387948563917844120
Offset: 1

Views

Author

Tom Richardson, Aug 29 2016

Keywords

Examples

			G.f.: A(x) = x+3*x^2+20*x^3+165*x^4+1524*x^5+15078*x^6+156264*x^7+...
Related Expansions:
A(x)^2 = x^2+6*x^3+49*x^4+450*x^5+4438*x^6+45900*x^7+491181*x^8+...
A(x)^3 = x^3+9*x^4+87*x^5+882*x^6+9282*x^7+100521*x^8+1113299*x^9+...

Links

Crossrefs

Cf. A250886, A276310, A276314, A276316.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)
  • PARI
    {a(n)=polcoeff(serreverse(x - 3*x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f.: Series_Reversion(x-3*x^2-2*x^3).
a(n) ~ (6*(18 + 5*sqrt(15))/17)^(n - 1/2) / (2*15^(1/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 22 2017
Showing 1-3 of 3 results.

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