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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A160150 Numerator of Hermite(n, 22/27).

Original entry on oeis.org

1, 44, 478, -107272, -6810740, 325937744, 63991555336, -35674949728, -654667511547248, -28389257894451520, 7341419739167121376, 736937848624456502144, -85316424437286206533952, -16647387274774084049005312, 884602468694263060488292480
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 44/27, 478/729, -107272/19683, -6810740/531441, ...

Links

Crossrefs

Cf. A009971 (denominators)

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(44/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],22/27]] (* Harvey P. Dale, Jul 29 2013 *)
Table[27^n*HermiteH[n, 22/27], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 22/27)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(44*x - 729*x^2))) \\ G. C. Greubel, Sep 24 2018

Formula

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 27^n * Hermite(n, 22/27).
E.g.f.: exp(44*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(44/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160151 Numerator of Hermite(n, 23/27).

Original entry on oeis.org

1, 46, 658, -103868, -7656020, 253581256, 67477123576, 885618857008, -647933055794288, -40134778914678560, 6655977728057433376, 891340052066655340096, -65746928407518970839872, -18619244257704074488953728, 389682045181727146807062400
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 46/27, 658/729, -103868/19683, -7656020/531441, ...

Links

Crossrefs

Cf. A009971 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(46/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018
  • Mathematica
    Table[27^n*HermiteH[n, 23/27], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)
HermiteH[Range[0,20],23/27]//Numerator (* Harvey P. Dale, Jan 02 2019 *)
  • PARI
    a(n)=numerator(polhermite(n, 23/27)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(46*x - 729*x^2))) \\ G. C. Greubel, Sep 24 2018

Formula

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 27^n * Hermite(n, 23/27).
E.g.f.: exp(46*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(46/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160152 Numerator of Hermite(n, 25/27).

Original entry on oeis.org

1, 50, 1042, -93700, -9242708, 84323000, 71595491320, 2842116962000, -588597736311920, -62580339060364000, 4594562542866814240, 1142149470643447832000, -16580120530325575181120, -20812053164894042027728000, -726343053712911149403451520
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 50/27, 1042/729, -93700/19683, -9242708/531441, ...

Links

Crossrefs

Cf. A009971 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(50/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018
  • Mathematica
    Numerator[HermiteH[Range[0,20],25/27]] (* Harvey P. Dale, Nov 15 2014 *)
Table[27^n*HermiteH[n, 25/27], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 25/27)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(50*x - 729*x^2))) \\ G. C. Greubel, Sep 24 2018

Formula

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 27^n * Hermite(n, 25/27).
E.g.f.: exp(50*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(50/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160153 Numerator of Hermite(n, 26/27).

Original entry on oeis.org

1, 52, 1246, -86840, -9965684, -11764688, 72038072584, 3848897264992, -535077911012720, -72717589071528128, 3239977716589449184, 1228701289925531463808, 11929704457466050105024, -20877013136748863885323520, -1311720301397752435727447936
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 52/27, 1246/729, -86840/19683, -9965684/531441, ...

Links

Crossrefs

Cf. A009971 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(52/27)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Sep 24 2018
  • Mathematica
    Table[27^n*HermiteH[n, 26/27], {n, 0, 30}] (* G. C. Greubel, Sep 24 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 26/27)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(52*x - 729*x^2))) \\ G. C. Greubel, Sep 24 2018

Formula

From G. C. Greubel, Sep 24 2018: (Start)
a(n) = 27^n * Hermite(n, 26/27).
E.g.f.: exp(52*x - 729*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(52/27)^(n-2*k)/(k!*(n-2*k)!)). (End)

A165848 Totally multiplicative sequence with a(p) = 27.

Original entry on oeis.org

1, 27, 27, 729, 27, 729, 27, 19683, 729, 729, 27, 19683, 27, 729, 729, 531441, 27, 19683, 27, 19683, 729, 729, 27, 531441, 729, 729, 19683, 19683, 27, 19683, 27, 14348907, 729, 729, 729, 531441, 27, 729, 729, 531441, 27, 19683, 27, 19683, 19683
Offset: 1

Views

Author

Jaroslav Krizek, Sep 28 2009

Keywords

Links

Crossrefs

Cf. A001222, A009971.

Programs

  • Magma
    [n eq 1 select 1 else 27^(&+[p[2]: p in Factorization(n)]): n in [1..100]]; // Vincenzo Librandi, Apr 14 2016
  • Mathematica
    27^PrimeOmega[Range[50]] (* Harvey P. Dale, Apr 20 2014 *)
  • PARI
    a(n) = 27^bigomega(n); \\ Altug Alkan, Apr 13 2016

Formula

a(n) = A009971(A001222(n)) = 27^bigomega(n) = 27^A001222(n).

A198080 a(n) = (3^(3*n + 3)- 26*n - 27)/169.

Original entry on oeis.org

0, 4, 116, 3144, 84904, 2292428, 61895580, 1671180688, 45121878608, 1218290722452, 32893849506244, 888133936668632, 23979616290053112, 647449639831434076, 17481140275448720108, 471990787437115442976, 12743751260802116960416, 344081284041657157931300
Offset: 0

Views

Author

Michel Lagneau, Oct 24 2011

Keywords

Comments

Second differences are four times the entries of A009971. - R. J. Mathar, Oct 25 2011

Examples

			a(1) = (3^(3 + 3) - 26 - 27)/169 = 676/169 = 4.

Links

Programs

  • Magma
    I:=[0, 4, 116]; [n le 3 select I[n] else 29*Self(n-1)-55*Self(n-2)+27*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Nov 25 2011
  • Maple
    for n from 0 to 30 do:x:=(3^(3*n+3) - 26*n - 27)/169  :  printf(`%d, `, x):od:
  • Mathematica
    LinearRecurrence[{29,-55,27},{0,4,116},50] (* Vincenzo Librandi, Nov 25 2011 *)
  • PARI
    a(n)=(3^(3*n+3)-26*n-27)/169 \\ Charles R Greathouse IV, Jul 06 2017

Formula

a(n) = (3^(3*n + 3) - 26*n - 27)/169.
G.f.: -4*x / ( (27*x-1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011

A306472 a(n) = 37*27^n.

Original entry on oeis.org

37, 999, 26973, 728271, 19663317, 530909559, 14334558093, 387033068511, 10449892849797, 282147106944519, 7617971887502013, 205685240962554351, 5553501505988967477, 149944540661702121879, 4048502597865957290733, 109309570142380846849791, 2951358393844282864944357
Offset: 0

Views

Author

Stefano Spezia, Feb 18 2019

Keywords

Comments

x = a(n) and y = A002042(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 3^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

Examples

			For a(0) = 37 and A002042(0) = 7, 37^2 + 3 = 1372 = 4*7^3.

Links

Crossrefs

Cf. A002042 (7*4^n), A009971 (27^n), A000290 (n^2), A000578 (n^3).

Programs

  • GAP
    List([0..20], n->37*27^n);
  • Magma
    [37*27^n: n in [0..20]];
  • Maple
    a:=n->37*27^n: seq(a(n), n=0..20);
  • Mathematica
    37*27^Range[0,20]
  • PARI
    a(n) = 37*27^n;

Formula

O.g.f.: 37/(1 - 27*x).
E.g.f.: 37*exp(27*x).
a(n) = 27*a(n-1) for n > 0.
a(n) = 37*A009971(n).

A381267 a(n) = numerator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).

Original entry on oeis.org

1, 15, 31185, 6381375, 409933148625, 115551955934415, 561860686475913825, 179982394552964750175, 245527483089290688069980625, 84259935283701238220954169375, 473788223464393905637179153328785, 169752647693877043154936308907932575, 15821279983229628402902553309640505635425
Offset: 0

Views

Author

Stefano Spezia, Feb 18 2025

Keywords

Links

Crossrefs

Cf. A381268 (denominator).
Cf. A009971, A134375.

Programs

  • Mathematica
    a[n_]:=Numerator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]

Formula

a(n) = numerator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).
a(n) = numerator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).

A381268 a(n) = denominator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).

Original entry on oeis.org

1, 4, 256, 1024, 1048576, 4194304, 268435456, 1073741824, 17592186044416, 70368744177664, 4503599627370496, 18014398509481984, 18446744073709551616, 73786976294838206464, 4722366482869645213696, 18889465931478580854784, 4951760157141521099596496896, 19807040628566084398385987584
Offset: 0

Views

Author

Stefano Spezia, Feb 18 2025

Keywords

Links

Crossrefs

Cf. A381267 (numerator).
Cf. A009971, A134375, A278142.

Programs

  • Mathematica
    a[n_]:=Denominator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]

Formula

a(n) = denominator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).
a(n) = denominator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).
a(2*n) = A278142(n).

A055156 Powers of 3 which are not powers of 3^3.

Original entry on oeis.org

3, 9, 81, 243, 2187, 6561, 59049, 177147, 1594323, 4782969, 43046721, 129140163, 1162261467, 3486784401, 31381059609, 94143178827, 847288609443, 2541865828329, 22876792454961, 68630377364883, 617673396283947
Offset: 0

Views

Author

Henry Bottomley, Jun 20 2000

Keywords

Links

Crossrefs

Cf. A013732 and A013733. Consists of numbers in A000244 which are not in A009971. See A004171 for powers of 2 which are not powers of 2^2.

Programs

  • Mathematica
    With[{nn=40},Complement[3^Range[nn],27^Range[Floor[nn/3]]]] (* or *) LinearRecurrence[{0,27},{3,9},40] (* Harvey P. Dale, Jul 17 2012 *)

Formula

a(n) = a(n-1)*a(n-2)/a(n-3) = 27*a(n-2) = 3^A001651(n).
a(2n) = 3^(3n+1), a(2n+1) = 3^(3n+2).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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