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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A159709 Numerator of Hermite(n, 5/21).

Original entry on oeis.org

1, 10, -782, -25460, 1814572, 107968600, -6922576520, -640595596400, 36334031470480, 4883382842903200, -239585713383638240, -45467293808242606400, 1869787653165632140480, 499923714198096067542400, -16439748089216177447319680, -6337455503810252016486752000
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerator of 1, 10/21, -782/441, -25460/9261, 1814572/194481, 107968600/4084101, ...

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(10/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 22 2018
  • Maple
    A159709 := proc(n)
        orthopoly[H](n,5/21) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 17 2014
  • Mathematica
    Numerator[Table[HermiteH[n, 5/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n,5/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) - 10*a(n-1) + 882*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 17 2014
From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,5/21).
E.g.f.: exp(10*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(10/21)^(n-2k)/(k!*(n-2k)!). (End)

A159745 Numerator of Hermite(n, 8/21).

Original entry on oeis.org

1, 16, -626, -38240, 1044556, 151623616, -2180514104, -837280401536, 66007653520, 5908906635694336, 94018537417467616, -50612259928144561664, -1721964008874583797056, 508128734937488699898880, 27874099084755797015426176, -5828388033652017714104551424
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(16/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 22 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 8/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n,8/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

D-finite with recurrence a(n) -16*a(n-1) +882*(n-1)*a(n-2)=0. [DLMF] - R. J. Mathar, Feb 17 2014
From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,8/21).
E.g.f.: exp(16*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(16/21)^(n-2k)/(k!*(n-2k)!). (End)

A159753 Numerator of Hermite(n, 10/21).

Original entry on oeis.org

1, 20, -482, -44920, 376972, 166017200, 1657897480, -845405072800, -27143960497520, 5422298983726400, 323914738103841760, -41346382274390012800, -3969548434571273011520, 358219141300718435244800, 52679225176808585054984320, -3369705453245099537303104000
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators)

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(20/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 14 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 10/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
Table[21^n*HermiteH[n, 10/21], {n,0,30}] (* G. C. Greubel, Jul 14 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 10/21)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(20*x - 441*x^2))) \\ G. C. Greubel, Jul 14 2018

Formula

From G. C. Greubel, Jul 14 2018: (Start)
a(n) = 21^n * Hermite(n, 10/21).
E.g.f.: exp(20*x - 441*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(20/21)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159761 Numerator of Hermite(n, 11/21).

Original entry on oeis.org

1, 22, -398, -47564, 6700, 167953192, 3665423224, -808168981136, -40410040569968, 4813419438356320, 426670129688245024, -33067616593161351872, -4867041163284902964032, 242912748429751883004544, 61149574443679238811690880, -1654195979849632997482909952
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(22/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 11/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n, 11/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,11/21).
E.g.f.: exp(22*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(22/21)^(n-2k)/(k!*(n-2k)!).
a(n+2) = 22*a(n+1) - 882*(n+1)*a(n). (End)

A159762 Numerator of Hermite(n, 13/21).

Original entry on oeis.org

1, 26, -206, -51220, -786644, 160251416, 7635636856, -649523935216, -64030044264560, 2918259736005536, 584145244508221216, -10551274514355075904, -5941710299591994211136, -42809778329457726121600, 67014596058555904745723776, 2270994640334597525538334976
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(26/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 13/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n, 13/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, May 22 2018: (Start)
a(n) = 21^n * Hermite(n,13/21).
E.g.f.: exp(26*x-441*x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*n!*(26/21)^(n-2k)/(k!*(n-2k)!).
a(n+2) = 26*a(n+1) - 882*(n+1)*a(n). (End)

A159763 Numerator of Hermite(n, 16/21).

Original entry on oeis.org

1, 32, 142, -51904, -2036660, 117944192, 12755884744, -215972352256, -85665947681648, -1217409408294400, 641059191631501024, 31251445113364640768, -5219510033581154430272, -497789616154448299657216, 43917634328099171108467840, 7552070478774301079638274048
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(32/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Maple
    seq(21^n*orthopoly[H](n,16/21), n=0..50); # Robert Israel, Jan 02 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 16/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
  • PARI
    a(n)=numerator(polhermite(n, 16/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From Robert Israel, Jan 02 2018: (Start)
a(n) = 21^n * Hermite(n,16/21).
E.g.f.: exp(32*x-441*x^2).
a(n+2) = 32*a(n+1) - 882*(n+1)*a(n). (End)

A159776 Numerator of Hermite(n, 17/21).

Original entry on oeis.org

1, 34, 274, -50660, -2447444, 95515384, 14040751096, -28081874864, -87642381012080, -2781695245370336, 601127582131299616, 44972889856630550464, -4303061546712430158656, -622297158830800371505280, 28180800294357511567970176, 8642272527250878380658183424
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators)

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(34/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 17/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
Table[21^n*HermiteH[n, 17/21], {n,0,50}] (* G. C. Greubel, Jul 11 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 17/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 21^n * Hermite(n, 17/21).
E.g.f.: exp(34*x - 441*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(37/21)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159784 Numerator of Hermite(n, 19/21).

Original entry on oeis.org

1, 38, 562, -45676, -3222740, 38680808, 15682154104, 391223020016, -81955144677488, -5874765126977440, 427318863624757024, 68053545237681787712, -1559812895855484713792, -779551612838132460267904, -11738146623970045761841280, 9179853743614397880438109952
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A009965 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(38/21)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, May 21 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 19/21], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 17 2011 *)
Table[21^n*HermiteH[n, 19/21], {n,0,50}] (* G. C. Greubel, Jul 11 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 19/21)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 21^n * Hermite(n, 19/21).
E.g.f.: exp(38*x - 441*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(38/21)^(n-2*k)/(k!*(n-2*k)!)). (End)

A165842 Totally multiplicative sequence with a(p) = 21.

Original entry on oeis.org

1, 21, 21, 441, 21, 441, 21, 9261, 441, 441, 21, 9261, 21, 441, 441, 194481, 21, 9261, 21, 9261, 441, 441, 21, 194481, 441, 441, 9261, 9261, 21, 9261, 21, 4084101, 441, 441, 441, 194481, 21, 441, 441, 194481, 21, 9261, 21, 9261, 9261, 441, 21, 4084101, 441
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2009

Keywords

Links

Programs

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

Formula

a(n) = A009965(A001222(n)) = 21^bigomega(n) = 21^A001222(n).

A178626 Convolution of Pell(n) and 10^n.

Original entry on oeis.org

1, 12, 125, 1262, 12649, 126560, 1265769, 12658098, 126581965, 1265822028, 12658226021, 126582274070, 1265822774161, 12658227822392, 126582278418945, 1265822784660282, 12658227847739509, 126582278480139300
Offset: 1

Views

Author

Mark Dols, May 31 2010

Keywords

Comments

Row sums of:
1
10....2
100...20...5
1000..200..50..12
10000.2000.500.120.29
or:
1
10....2
100...21...4
1000..210..44..8
10000.2100.441.92.16
or without the first column:
1
10
100......1
1000.....12
10000....124....1
100000...1248...14

Links

Crossrefs

Cf. A021083, A000129, A038207, A009965.

Programs

  • GAP
    a:=[1,12,125];; for n in [4..30] do a[n]:=12*a[n-1]-19*a[n-2] - 10*a[n-3]; od; a; # G. C. Greubel, Jan 28 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x/((10*x-1)*(x^2+2*x-1)) )); // G. C. Greubel, Jan 28 2019
  • Mathematica
    LinearRecurrence[{12, -19, -10}, {1, 12, 125}, 30] (* G. C. Greubel, Jan 28 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec( x/((10*x-1)*(x^2+2*x-1)) ) \\ G. C. Greubel, Jan 28 2019
  • Sage
    a=(x/((10*x-1)*(x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 28 2019

Formula

a(n) = 10*a(n-1) + Pell(n) with a(1)= Pell(1) = 1.
From R. J. Mathar, May 31 2010: (Start)
a(n) = +12*a(n-1) -19*a(n-2) -10*a(n-3).
G.f.: x/ ( (10*x-1)*(x^2+2*x-1) ). (End)
a(n) = (2^(3+n)*5^(1+n) + (1-sqrt(2))^n*(-20+11*sqrt(2)) - (1+sqrt(2))^n*(20+11*sqrt(2))) / 316. - Colin Barker, Jan 29 2019

Extensions

a(10), a(11) corrected and sequence extended by R. J. Mathar, May 31 2010
Previous Showing 21-30 of 32 results. Next

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

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