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.

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A159949 Numerator of Hermite(n, 1/24).

Original entry on oeis.org

1, 1, -287, -863, 247105, 1241281, -354589919, -2499523487, 712353753217, 6471255867265, -1839949672471199, -20477166570194399, 5808483395818564033, 76577571062410406977, -21670384262882293332575, -330431150786521054263839, 93285628864864986142460161
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 1/12, -287/144, -863/1728, 247105/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(1/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],1/24]] (* Harvey P. Dale, Nov 05 2016 *)
Table[12^n*HermiteH[n, 1/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 1/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 1/24).
E.g.f.: exp(x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159954 Numerator of Hermite(n, 5/24).

Original entry on oeis.org

1, 5, -263, -4195, 206257, 5863925, -267690455, -11471314675, 482307383905, 28841445930725, -1105933509428135, -88593031827628675, 3060632198730188305, 321480678989935642325, -9851603557096146802295, -1345468115472901243865875, 35831586789290847966585025
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 5/12, -263/144, -4195/1728, 206257/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(5/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],5/24]] (* Harvey P. Dale, Feb 22 2016 *)
Table[12^n*HermiteH[n, 5/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 5/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(5*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 5/24).
E.g.f.: exp(5*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(5/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159967 Numerator of Hermite(n, 7/24).

Original entry on oeis.org

1, 7, -239, -5705, 166561, 7738087, -185681231, -14671182953, 271635081025, 35703851090887, -454151172380591, -106006149348418697, 696707868662781409, 371234207228774486695, -9834809672032188431, -1496885167214122955673257, -10435709792715681635690879
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 7/12, -239/144, -5705/1728, 166561/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(7/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],7/24]] (* Harvey P. Dale, Jan 27 2012 *)
Table[12^n*HermiteH[n, 7/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 7/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(7*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 7/24).
E.g.f.: exp(7*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(7/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159968 Numerator of Hermite(n, 11/24).

Original entry on oeis.org

1, 11, -167, -8173, 54385, 10013531, 31834441, -16953202717, -250663462943, 36302880967595, 1049051386591801, -93012731934163789, -4346534843998627247, 273640118280485155067, 19283467757016197118505, -891198811579737976926589, -93107767637687089298134079
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 11/12, -167/144, -8173/1728, 54385/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(11/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],11/24]] (* Harvey P. Dale, Mar 27 2013 *)
Table[12^n*HermiteH[n, 11/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 11/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(11*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 11/24).
E.g.f.: exp(11*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(11/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159969 Numerator of Hermite(n, 13/24).

Original entry on oeis.org

1, 13, -119, -9035, -14639, 10218013, 153914329, -15655840187, -513817209695, 29391432064813, 1713902824372009, -62366587629825323, -6240409786798253711, 134413599620299018045, 25111471036836549128569, -215506510190170502086043, -111283139511606108762536639
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 13/12, -119/144, -9035/1728, -14639/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(13/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 13/24], {n, 0, 30}]] (* or *) Table[12^n* HermiteH[n, 1/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 13/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(13*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 13/24).
E.g.f.: exp(13*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(13/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159996 Numerator of Hermite(n, 17/24).

Original entry on oeis.org

1, 17, 1, -9775, -167039, 8421137, 383695489, -8028901423, -910021430015, 3028224568337, 2410255364260609, 32253054435619793, -7087387068572072831, -231952136295227242735, 22591990867714977552769, 1319294858293510861104593, -75169387957539018389183999
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 17/12, 1/144, -9775/1728, -167039/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(17/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 17/24], {n, 0, 30}]] (* or *) Table[12^n* HermiteH[n, 1/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 17/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(17*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 17/24).
E.g.f.: exp(17*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(17/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159997 Numerator of Hermite(n, 19/24).

Original entry on oeis.org

1, 19, 73, -9557, -244655, 6361219, 473166361, -2002025573, -991941869663, -14234228603405, 2300662982701801, 84707175049140619, -5679064003265633807, -400650213031877021597, 13650061580620869563065, 1874772828976324672777339, -23347582277731987729671359
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 19/12, 73/144, -9557/1728, -244655/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(19/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],19/24]] (* Harvey P. Dale, Jun 12 2016 *)
Table[12^n*HermiteH[n, 19/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 19/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(19*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 19/24).
E.g.f.: exp(19*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(19/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159998 Numerator of Hermite(n, 23/24).

Original entry on oeis.org

1, 23, 241, -7705, -385439, 11063, 555286609, 12752475143, -826150875455, -48383172864937, 1028570093285809, 163000649996592167, 490504894392176929, -552048633817202459785, -14533568902399966997231, 1891588006795761076916807, 106291541814670362197124481
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 23/12, 241/144, -7705/1728, -385439/20736, ...

Links

Crossrefs

Cf. A001021 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(23/12)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 16 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 23/24], {n, 0, 30}]] (* or *) Table[12^n* HermiteH[n, 1/12], {n, 0, 30}] (* G. C. Greubel, Jul 16 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 23/24)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(23*x - 144*x^2))) \\ G. C. Greubel, Jul 16 2018

Formula

From G. C. Greubel, Jul 16 2018: (Start)
a(n) = 12^n * Hermite(n, 23/24).
E.g.f.: exp(23*x - 144*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(23/12)^(n-2*k)/(k!*(n-2*k)!)). (End)

A165833 Totally multiplicative sequence with a(p) = 12.

Original entry on oeis.org

1, 12, 12, 144, 12, 144, 12, 1728, 144, 144, 12, 1728, 12, 144, 144, 20736, 12, 1728, 12, 1728, 144, 144, 12, 20736, 144, 144, 1728, 1728, 12, 1728, 12, 248832, 144, 144, 144, 20736, 12, 144, 144, 20736, 12, 1728, 12, 1728, 1728, 144, 12, 248832, 144, 1728
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2009

Keywords

Links

Crossrefs

Cf. A001021, A001222.

Programs

  • Mathematica
    12^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

Formula

a(n) = A001021(A001222(n)) = 12^bigomega(n) = 12^A001222(n).
Dirichlet g.f.: Product_{p prime} 1 / (1 - 12 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A176771 Smallest power of 12 whose decimal expansion contains n.

Original entry on oeis.org

20736, 1, 12, 20736, 144, 2985984, 20736, 1728, 1728, 2985984, 8916100448256, 2218611106740436992, 12, 79496847203390844133441536, 144, 5159780352, 429981696, 1728, 35831808, 61917364224, 20736, 15407021574586368
Offset: 0

Views

Author

Jonathan Vos Post, Apr 25 2010

Keywords

Comments

This is to 12 as A176763 is to 3 and as A030001 is to 2.

Examples

			a(1) = 1 because 12^0 = 1 has "1" as a substring (not a proper substring, though).
a(2) = 12 because 12^1 = 12 has "2" as a substring.
a(3) = 20736 because 12^4 = 20736 has "3" as a substring.

Links

Crossrefs

Cf. A001018, A030001, A176763, A176764-A176773.

Programs

  • Mathematica
    A176771[n_] := Block[{k = -1}, While[StringFreeQ[IntegerString[12^++k], IntegerString[n]]]; 12^k]; Array[A176771, 50, 0] (* Paolo Xausa, Apr 04 2024 *)

Formula

a(n) = MIN{A001021(i) such that n in decimal representation is a substring of A001021(i)}.

Extensions

More terms from Sean A. Irvine and Jon E. Schoenfield, May 05 2010
a(0) prepended by Paolo Xausa, Apr 04 2024
Previous Showing 41-50 of 58 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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