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 1-10 of 10 results.

A275452 G.f.: 3F2([1/9, 4/9, 7/9], [1/3, 1], 729 x).

Original entry on oeis.org

1, 84, 32760, 16302000, 9020711700, 5299182393120, 3234930051733380, 2028415806982164600, 1297264109283593451000, 842341453312777393815840, 553562736218491223861661024, 367351669654325623384052435136, 245756466255265144369306647476400
Offset: 0

Views

Author

Gheorghe Coserea, Jul 30 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 84*x + 32760*x^2 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 7/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2016 *)
FullSimplify[Table[(729^n Gamma[1/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] Gamma[1/3 + n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 09 2016 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x, N)

Formula

G.f.: hypergeom([1/9, 4/9, 7/9], [1/3, 1], 729*x).
From Vaclav Kotesovec, Jul 31 2016: (Start)
Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 2)*a(n-1).
a(n) ~ Gamma(1/3) * 3^(6*n) / (Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * n).
a(n) ~ 2^(2/9) * Gamma(1/3) * sin(Pi/9) * 3^(6*n) / (sqrt(Pi) * Gamma(4/9) * Gamma(7/18) * n).
(End)
a(n) = (729^n * Gamma(1/3) * Gamma(1/9 + n) * Gamma(4/9+n) * Gamma(7/9 + n))/(n!^2*Gamma(1/9) * Gamma(4/9) * Gamma(7/9) * Gamma(1/3 + n)). - Benedict W. J. Irwin, Aug 09 2016

A275453 G.f.: 3F2([1/9, 4/9, 7/9], [2/3, 1], 729 x).

Original entry on oeis.org

1, 42, 13104, 5705700, 2870226450, 1565667525240, 899552741658480, 535848881630582520, 327799728893143306800, 204660966917426732512800, 129859500691523648952466560, 83483493583251639541209993720, 54254332317972702411364923299700, 35581785531539194815959254026276000
Offset: 0

Views

Author

Gheorghe Coserea, Jul 30 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 42*x + 13104*x^2 + 5705700*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    a[n_] := FullSimplify[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[7/9 + n])/((n!)^2 Gamma[1/9] Gamma[4/9] Gamma[7/9] Gamma[2/3 + n])] (* Benedict W. J. Irwin, Aug 05 2016 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 4/9, 7/9], [2/3, 1], 729*x, N))
  • PARI
    a(n) = round(729^n*gamma(2/3)*gamma(1/9+n)*gamma(4/9+n)*gamma(7/9+n)/((n!)^2*gamma(1/9)*gamma(4/9)*gamma(7/9)*gamma(2/3+n))) \\ Charles R Greathouse IV, Aug 05 2016

Formula

G.f.: hypergeom([1/9, 4/9, 7/9], [2/3, 1], 729*x).
a(n) = 729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(7/9+n)/((n!)^2*Gamma(1/9)*Gamma(4/9)*Gamma(7/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 05 2016
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-8)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A275454 G.f.: 3F2([1/9, 4/9, 8/9], [2/3, 1], 729 x).

Original entry on oeis.org

1, 48, 15912, 7205484, 3731294385, 2082701917296, 1219626159039288, 738421413473848104, 458174434421099404008, 289681112497807349679360, 185894363292170517130962816, 120738965077159251405022531728, 79206198459248339865163888224660, 52397749335891513408552541281755520
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 48*x + 15912*x^2 + 7205484*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    FullSimplify[Table[(729^n Gamma[2/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[8/9 + n] Sin[Pi/9])/(Pi (n!)^2 Gamma[4/9] Gamma[2/3 + n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 09 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 8/9}, {2/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 10 2016 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x, N))

Formula

G.f.: hypergeom([1/9, 4/9, 8/9], [2/3, 1], 729*x).
a(n) = (729^n*Gamma(2/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(4/9)*Gamma(2/3+n)). - Benedict W. J. Irwin, Aug 09 2016
a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(1/3)*Gamma(4/9)*n^(11/9)). - Vaclav Kotesovec, Aug 10 2016
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-8)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A275455 G.f.: 3F2([1/9, 5/9, 8/9], [1/3, 1], 729 x).

Original entry on oeis.org

1, 120, 53550, 28973100, 17036182800, 10496595041856, 6664244456261700, 4320449008019199000, 2844426519643185378000, 1894935877560218667820800, 1274265873172890987907535424, 863426385292565961502380501120, 588738285265666300220495724048000, 403569219885941102398195162309056000
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 120*x + 53550*x^2 + 28973100*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    FullSimplify[Table[(729^n Gamma[1/3]Gamma[1/9+n]Gamma[5/9+n]Gamma[8/9+n]Sin[Pi/9])/(Pi n!^2Gamma[5/9]Gamma[1/3+n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 10 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/9, 5/9, 8/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 13 2016 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 5/9, 8/9], [1/3, 1], 729*x, N))

Formula

G.f.: hypergeom([1/9, 5/9, 8/9], [1/3, 1], 729*x).
a(n) = (729^n*Gamma(1/3)*Gamma(1/9+n)*Gamma(5/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(5/9)*Gamma(1/3+n)). - Benedict W. J. Irwin, Aug 10 2016
a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(2/3)*Gamma(5/9)*n^(7/9)). - Vaclav Kotesovec, Aug 13 2016
D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-4)*(9*n-8)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A275456 G.f.: 3F2([1/9, 7/9, 8/9], [1/3, 1], 729 x).

Original entry on oeis.org

1, 168, 85680, 50388000, 31479903000, 20342022734880, 13431668094985140, 9002968680250888200, 6101557410115488321000, 4170391891453158061891200, 2869634745103513910507157888, 1985363415926004500849300108544, 1379778913200535726019164327886400, 962553011288199733460143650698784000
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 168*x + 85680*x^2 + 50388000*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    FullSimplify[Table[(729^n Gamma[1/3]Gamma[1/9+n]Gamma[7/9+n]Gamma[8/9+n]Sin[Pi/9]) / (Pi n!^2Gamma[7/9]Gamma[1/3+n]), {n, 0, 20}]] (* Benedict W. J. Irwin, Aug 10 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/9, 7/9, 8/9}, {1/3, 1}, 729*x], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 13 2016 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x, N))

Formula

G.f.: hypergeom([1/9, 7/9, 8/9], [1/3, 1], 729*x).
a(n) = (729^n*Gamma(1/3)*Gamma(1/9+n)*Gamma(7/9+n)*Gamma(8/9+n)*sin(Pi/9)) / (Pi*n!^2*Gamma(7/9)*Gamma(1/3+n)). - Benedict W. J. Irwin, Aug 10 2016
a(n) ~ 2*sin(Pi/9)*3^(6*n-1/2) / (Gamma(2/3)*Gamma(7/9)*n^(5/9)). - Vaclav Kotesovec, Aug 13 2016
D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-8)*(9*n-2)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022

A275457 G.f.: 3F2([2/9, 4/9, 5/9], [1/3, 1], 729 x).

Original entry on oeis.org

1, 120, 45045, 21707400, 11708971560, 6735720993408, 4039678502036100, 2494516661768577600, 1573990406710539567750, 1009797626141015909237040, 656436978973434195655059942, 431326871057383042747830748560, 285942228994752084893009228453460, 190985447073724962020463006948873600
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 120*x + 45045*x^2 + 21707400*x^3 + ...

Links

Crossrefs

Cf. A007949, A053735, A268545-A268555, A275051-A275054.

Programs

  • Maple
    A[0]:= 1:
for n from 0 to 20 do A[n+1]:= 3*(5+9*n)*(2+9*n)*(4+9*n)*A[n]/((n+1)^2*(3*n+1)) od:
seq(A[i],i=0..21); # Robert Israel, Jan 20 2017
  • Mathematica
    CoefficientList[HypergeometricPFQ[{2/9, 4/9, 5/9}, {1/3, 1}, 729 x] + O[x]^14, x] (* Jean-François Alcover, Sep 18 2018 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([2/9, 4/9, 5/9], [1/3, 1], 729*x, N))

Formula

G.f.: hypergeom([2/9, 4/9, 5/9], [1/3, 1], 729*x).
From Robert Israel, Jan 20 2017: (Start)
a(n) = (2/3)*729^n*Gamma(5/9+n)*Gamma(2/9+n)*Gamma(4/9+n)*sin((4/9)*Pi)*3^(1/2)/(Gamma(2/9)*Gamma(n+1)^2*Gamma(n+1/3)*Gamma(2/3)).
D-finite with recurrence a(n+1) = 3*(5+9*n)*(2+9*n)*(4+9*n)*a(n)/((n+1)^2*(3*n+1)).
a(n) ~ (2*sin(4*Pi/9)/(sqrt(3)*Gamma(2/9)*Gamma(2/3)))*729^n/n^(10/9).
A007949(a(n)) = A053735(n). (End)

A275458 G.f.: 3F2([4/9, 5/9, 7/9], [2/3, 1], 729 x).

Original entry on oeis.org

1, 210, 91728, 48348300, 27795877200, 16801416515520, 10492649333712000, 6704867164952174400, 4357981459741604877000, 2869985317222538272758000, 1909866367099566641482516800, 1281775836140482143996826609500, 866321769175062822028788514251300, 589012467640059218480339437176228000
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 210*x + 91728*x^2 + 48348300*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    HypergeometricPFQ[{4/9, 5/9, 7/9}, {2/3, 1}, 729 x] + O[x]^14 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2018 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([4/9, 5/9, 7/9], [2/3, 1], 729*x, N))

Formula

G.f.: hypergeom([4/9, 5/9, 7/9], [2/3, 1], 729*x).
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ (1 + 2*cos(2*Pi/9)) * Gamma(2/9) * 3^(6*n - 1/2) / (2*Pi*Gamma(1/3) * n^(8/9)). - Vaclav Kotesovec, Apr 27 2024

A275459 G.f.: 3F2([4/9, 5/9, 8/9], [2/3, 1], 729 x).

Original entry on oeis.org

1, 240, 111384, 61056996, 36134640360, 22349791271808, 14226080375707200, 9239577908667986880, 6091267058935364926620, 4062233028933305475849600, 2733980882372812975378956480, 1853783080629966591378982417800, 1264747920529034302126861656883140, 867379957865303554725274256161714560
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 240*x + 111384*x^2 + 61056996*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    HypergeometricPFQ[{4/9, 5/9, 8/9}, {2/3, 1}, 729 x] + O[x]^14 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2018 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([4/9, 5/9, 8/9], [2/3, 1], 729*x, N))

Formula

G.f.: hypergeom([4/9, 5/9, 8/9], [2/3, 1], 729*x).
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-5)*(9*n-4)*(9*n-1)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ Gamma(1/9) * (1 + 2*sin(Pi/18)) * 3^(6*n - 1/2) / (2*Pi*Gamma(1/3) * n^(7/9)). - Vaclav Kotesovec, Apr 27 2024

A275460 G.f.: 3F2([2/9, 4/9, 7/9], [1/3, 1], 729 x).

Original entry on oeis.org

1, 168, 72072, 37752000, 21636143100, 13053584427840, 8141901337189620, 5198083656717631680, 3376354693360163389875, 2222371681246143931063560, 1478289894198059998030179204, 991793399749992922720024531872, 670139971927397485144595595426978, 455519420546971097210713116712430400
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 168*x + 72072*x^2 + 37752000*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    HypergeometricPFQ[{2/9, 4/9, 7/9}, {1/3, 1}, 729 x] + O[x]^14 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2018 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([2/9, 4/9, 7/9], [1/3, 1], 729*x, N))

Formula

G.f.: hypergeom([2/9, 4/9, 7/9], [1/3, 1], 729*x).
D-finite with recurrence n^2*(3*n-2)*a(n) -3*(9*n-7)*(9*n-5)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ Gamma(1/3) * sin(2*Pi/9) * 3^(6*n) / (Pi * Gamma(4/9) * n^(8/9)). - Vaclav Kotesovec, Apr 27 2024

A275461 G.f.: 3F2([2/9, 5/9, 7/9], [2/3, 1], 729 x).

Original entry on oeis.org

1, 105, 38808, 18595500, 10000998000, 5742915942960, 3440119256028000, 2122455291847675200, 1338358017590361495000, 858192528139829777895000, 557657055926757140695941600, 366299456771890110076863664500, 242765837117133913048941576656100, 162109136966873437562041203714292500
Offset: 0

Views

Author

Gheorghe Coserea, Jul 31 2016

Keywords

Comments

"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).

Examples

			1 + 105*x + 38808*x^2 + 18595500*x^3 + ...

Links

Crossrefs

Cf. A268545-A268555, A275051-A275054.

Programs

  • Mathematica
    HypergeometricPFQ[{2/9, 5/9, 7/9}, {2/3, 1}, 729 x] + O[x]^14 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 23 2018 *)
  • PARI
    \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([2/9, 5/9, 7/9], [2/3, 1], 729*x, N))

Formula

G.f.: hypergeom([2/9, 5/9, 7/9], [2/3, 1], 729*x).
D-finite with recurrence n^2*(3*n-1)*a(n) -3*(9*n-7)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
a(n) ~ (1 + 2*cos(2*Pi/9)) * Gamma(4/9) * 3^(6*n - 1/2) / (2*Pi * Gamma(1/3) * n^(10/9)). - Vaclav Kotesovec, Apr 27 2024
Showing 1-10 of 10 results.

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