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 1-9 of 9 results.

A158968 Numerator of Hermite(n, 1/6).

Original entry on oeis.org

1, 1, -17, -53, 865, 4681, -73169, -578717, 8640577, 91975825, -1307797649, -17863446149, 241080488353, 4099584856537, -52313249418065, -1085408633265389, 13039168709612161, 325636855090044193, -3664348770051277073, -109170689819225595605, 1144036589538311163361
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158811, A158960.
Sequences with e.g.f = exp(x + q*x^2): this sequence (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), A115329 (q=2), A293720 (q=4).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(1/3)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 10 2018
  • Mathematica
    Numerator[Table[HermiteH[n,1/6],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011 *)
Table[3^n*HermiteH[n, 1/6], {n,0, 50}] (* G. C. Greubel, Jul 10 2018 *)
  • PARI
    a(n)=numerator(polhermite(n,1/6)) \\ Charles R Greathouse IV, Jan 29 2016
  • SageMath
    [3^n*hermite(n, 1/6) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 3^n * Hermite(n, 1/6).
E.g.f.: exp(x - 9*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/3)^(n-2*k)/(k!*(n-2*k)!)). (End)
D-finite with recurrence a(n) -a(n-1) +18*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021

A158980 Numerator of Hermite(n, 1/7).

Original entry on oeis.org

1, 2, -94, -580, 26476, 280312, -12412616, -189648688, 8135757200, 164956085792, -6845825678816, -175348615433792, 7029102850896064, 220268177451931520, -8514540677137722496, -319237020818325490432, 11877900753755801088256, 524319450150645971173888
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158811, A158954, A158960.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(2/7)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018
  • Mathematica
    Numerator[Table[HermiteH[n,1/7],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*)
  • PARI
    a(n)=numerator(polhermite(n,1/7)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jun 09 2018: (Start)
a(n) = 7^n * Hermite(n,1/7).
E.g.f.: exp(2*x-49*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/7)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159030 Numerator of Hermite(n, 1/9).

Original entry on oeis.org

1, 2, -158, -964, 74860, 774392, -59087816, -870884656, 65263814032, 1259194142240, -92636252574176, -2225167015577152, 160627468056027328, 4646979614394038144, -328987488497205476480, -11197324742440089463552, 777044947563329128919296
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158811, A158954, A158960, A158980.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(2/9)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018
  • Mathematica
    Numerator[Table[HermiteH[n,1/9],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*)
  • PARI
    a(n)=numerator(polhermite(n,1/9)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    for(n=0,30, print1(9^n*polhermite(n,1/9), ", ")) \\ G. C. Greubel, Jun 10 2018

Formula

From G. C. Greubel, Jun 09 2018: (Start)
a(n) = 9^n * Hermite(n, 1/9).
E.g.f.: exp(2*x-81*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/9)^(n-2k)/(k!*(n-2k)!)). (End)

A159014 Numerator of Hermite(n, 1/8).

Original entry on oeis.org

1, 1, -31, -95, 2881, 15041, -445919, -3333791, 96552065, 950002561, -26856992159, -330857811679, 9122803428289, 136172203113025, -3658914023055199, -64664061017690399, 1691614670048805121, 34799613911106289409, -885438766595443696415
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158811, A158954, A158960, A158980.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(1/4)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 10 2018
  • Mathematica
    Numerator[Table[HermiteH[n,1/8],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*)
  • PARI
    a(n)=numerator(polhermite(n,1/8)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jun 10 2018: (Start)
a(n) = 4^n * Hermite(n,1/8).
E.g.f.: exp(x-16*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/4)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159247 Numerator of Hermite(n, 1/10).

Original entry on oeis.org

1, 1, -49, -149, 7201, 37001, -1763249, -12863549, 604273601, 5749693201, -266173427249, -3141020027749, 143254364959201, 2027866381608601, -91087470841872049, -1510593937967892749, 66805009193436144001, 1275280159567750343201, -55508977654852972057649
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158811, A158954, A158960.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(1/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 10 2018
  • Mathematica
    Numerator[Table[HermiteH[n,1/10],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)
  • PARI
    a(n)=numerator(polhermite(n,1/10)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jun 10 2018: (Start)
a(n) = 5^n * Hermite(n,1/10).
E.g.f.: exp(x-25*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(1/5)^(n-2*k)/(k!*(n-2*k)!)). (End)
a(n) = 50*(1-n)*a(n-2)+a(n-1) for n>1. - Christian Krause, Oct 21 2024

A158961 Numerator of Hermite(n, 2/5).

Original entry on oeis.org

1, 4, -34, -536, 2956, 119024, -262904, -36758816, -55018864, 14483450944, 82692292576, -6910956301696, -73124586123584, 3854075436523264, 62947282726422656, -2446063674660594176, -56994716743459368704, 1728872072754637865984
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158954, A158960.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(4/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 09 2018
  • Mathematica
    Numerator[Table[HermiteH[n,2/5],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011 *)
Table[5^n*HermiteH[n, 2/5], {n,0,30}] (* G. C. Greubel, Jul 09 2018 *)
  • PARI
    a(n)=numerator(polhermite(n,2/5)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From G. C. Greubel, Jul 09 2018: (Start)
a(n) = 5^n * Hermite(n, 2/5).
E.g.f.: exp(4*x-25*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/5)^(n-2*k)/(k!*(n-2*k)!)). (End)

A237987 a(n) = Hermite(n,1/5)*5^n/2^round(n/2)*(-1)^floor(n/2).

Original entry on oeis.org

1, 1, 23, 73, 1579, 8879, 179617, 1511467, 28410041, 330703441, 5730852343, 88406712593, 1399170969139, 27921184747039, 398888195476097, 10171302856939747, 129240467589656881, 4197761610365555681, 46531675504873063063, 1935524400169373119513
Offset: 0

Views

Author

M. F. Hasler, Feb 16 2014

Keywords

Comments

The "quintessence" of sequence A158960. Intended to be read as: H_n(1/5) = (-1)^floor(n/2)*2^round(n/2)*a(n)/5^n; where "round" could be replaced by "ceiling".
First negative term is a(32). Georg Fischer, Feb 15 2019

Links

Programs

  • Mathematica
    Table[HermiteH[n, 1/5]*5^n/2^Ceiling[n/2]*(-1)^Floor[n/2], {n, 0, 50}] (* G. C. Greubel, Jul 12 2018 *)
  • PARI
    vector(30,n,polhermite(n,1/5)*5^n/2^((n+1)\2)/(-1)^(n\2))

A158965 Numerator of Hermite(n, 3/5).

Original entry on oeis.org

1, 6, -14, -684, -2004, 124776, 1249656, -29934864, -616988784, 8272012896, 327277030176, -2172344266944, -193036432198464, 145187966975616, 126344808730855296, 656437275502200576, -90819982895128268544, -1070069717772530072064, 70776567154223847830016
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158960, A158961.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(6/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 13 2018
  • Mathematica
    Numerator[Table[HermiteH[n,3/5],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*)
Table[5^n*HermiteH[n, 3/5], {n,0,30}] (* G. C. Greubel, Jul 13 2018 *)
  • PARI
    a(n)=numerator(polhermite(n,3/5)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(6*x - 25*x^2))) \\ G. C. Greubel, Jul 13 2018

Formula

From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 5^n * Hermite(n, 3/5).
E.g.f.: exp(6*x - 25*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(6/5)^(n-2*k)/(k!*(n-2*k)!)). (End)

A158967 Numerator of Hermite(n, 4/5).

Original entry on oeis.org

1, 8, 14, -688, -7604, 76768, 2515144, -2909248, -903574384, -6064895872, 358089305824, 5897162382592, -149771819142464, -4736471982694912, 59459906581042304, 3791209640534776832, -14265252811503513344, -3147089734919849572352
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Links

Crossrefs

Cf. A158960, A158961.

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(8/5)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 13 2018
  • Mathematica
    Numerator[Table[HermiteH[n,4/5],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 01 2011*)
Table[5^n*HermiteH[n, 4/5], {n,0,30}] (* G. C. Greubel, Jul 14 2018 *)
  • PARI
    a(n)=numerator(polhermite(n,4/5)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(8*x - 25*x^2))) \\ G. C. Greubel, Jul 14 2018

Formula

From G. C. Greubel, Jul 14 2018: (Start)
a(n) = 5^n * Hermite(n, 4/5).
E.g.f.: exp(8*x - 25*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(8/5)^(n-2*k)/(k!*(n-2*k)!)). (End)
D-finite with recurrence a(n) -8*a(n-1) +50*(n-1)*a(n-2)=0. - [DLMF] Georg Fischer, Feb 06 2021
Showing 1-9 of 9 results.

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

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