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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A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717
Offset: 0

Views

Author

Omar E. Pol, Sep 09 2018

Keywords

Comments

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

Examples

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Crossrefs

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

Programs

  • PARI
    T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

Formula

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A086874 Seventh power of odd primes.

Original entry on oeis.org

2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, 17249876309, 27512614111, 94931877133, 194754273881, 271818611107, 506623120463, 1174711139837, 2488651484819, 3142742836021, 6060711605323
Offset: 1

Views

Author

Douglas Winston (douglas.winston(AT)srupc.com), Sep 16 2003

Keywords

Crossrefs

Cf. A000040, A001248, A030078, A030514, A050997, A030516, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.

Programs

A160299 Numerator of Hermite(n, 1/31).

Original entry on oeis.org

1, 2, -1918, -11524, 11036140, 110668792, -105835967816, -1487904444976, 1420941302106512, 25719901350164000, -24528002841138116576, -543392509632428313152, 517484251048077204023488, 13567773344258481022584704, -12902725949998740057685701760
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 2/31, -1918/961, -11524/29791, 11036140/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(2/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Table[31^n*HermiteH[n, 1/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 1/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 1/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(2*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 2*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 1/31).
E.g.f.: exp(2*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160300 Numerator of Hermite(n, 2/31).

Original entry on oeis.org

1, 4, -1906, -23000, 10897996, 220415984, -103848077624, -2957229437984, 1385343118601360, 51011732312847424, -23759618336314935584, -1075483968398187231616, 498023914992777619190464, 26797057907106900786753280, -12336437308381113989945920384
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 4/31, -1906/961, -23000/29791, 10897996/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(4/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Table[31^n*HermiteH[n, 2/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 2/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 2/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(4*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 4*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 2/31).
E.g.f.: exp(4*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160301 Numerator of Hermite(n, 3/31).

Original entry on oeis.org

1, 6, -1886, -34380, 10668396, 328323816, -100553342664, -4389550302096, 1326507370388880, 75452769667361376, -22493207874982677984, -1585161480256581714624, 466040432011344287649984, 39356406972705866391987840, -11408347792399213172870573184
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 6/31, -1886/961, -34380/29791, 10668396/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(6/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Table[31^n*HermiteH[n, 3/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 3/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 3/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(6*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 6*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 3/31).
E.g.f.: exp(6*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(6/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160302 Numerator of Hermite(n, 4/31).

Original entry on oeis.org

1, 8, -1858, -45616, 10348300, 433482208, -95979305336, -5766751265344, 1245171563867792, 98630939966871680, -20749930192050092576, -2061686107699674430208, 422201535258725661800128, 50928340670055096352718336, -10141700834614078614916251520
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 8/31, -1858/961, -45616/29791, 10348300/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • GAP
    List(List([0..15],n->Sum([0..Int(n/2)],k->(-1)^k*Factorial(n)*(8/31)^(n-2*k)/(Factorial(k)*Factorial(n-2*k)))),NumeratorRat); # Muniru A Asiru, Jul 12 2018
  • Magma
    I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-1922*(n-2)*Self(n-2): n in [1..15]]; // Vincenzo Librandi, Mar 28 2018
  • Maple
    seq(orthopoly[H](n,4/31)*31^n, n=0..40); # Robert Israel, Mar 27 2018
  • Mathematica
    Numerator[Table[HermiteH[n, 4/31], {n, 0, 40}]] (* Vincenzo Librandi, Mar 28 2018 *)
  • PARI
    a(n)=numerator(polhermite(n, 4/31)) \\ Charles R Greathouse IV, Jan 29 2016

Formula

From Robert Israel, Mar 27 2018: (Start)
a(n+2) = 8*a(n+1) - 1922*(n+1)*a(n).
E.g.f.: exp(-961*x^2+8*x). (End)
a(n) = 31^n * Hermite(n, 4/31). - G. C. Greubel, Jul 12 2018

A160303 Numerator of Hermite(n, 5/31).

Original entry on oeis.org

1, 10, -1822, -56660, 9939052, 534992600, -90164363720, -7071178300400, 1142359566484880, 120150033211799200, -18559035448937462240, -2494873992820155246400, 367426387533234274214080, 61216037645736403345110400, -8568355342448027542061898880
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 10/31, -1822/961, -56660/29791, 9939052/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(10/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Numerator/@HermiteH[Range[0, 20], 5/31] (* Harvey P. Dale, May 14 2011 *)
Table[31^n*HermiteH[n, 5/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 5/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 5/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(10*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 10*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 5/31).
E.g.f.: exp(10*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(10/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160304 Numerator of Hermite(n, 6/31).

Original entry on oeis.org

1, 12, -1778, -67464, 9442380, 631971792, -83157610296, -8285790028896, 1019373008575632, 139634783587212480, -15957496899294732576, -2875270503337760656512, 302870153404836108243648, 69949680729840145080716544, -6728117484215153259607190400
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 12/31, -1778/961, -67464/29791, 9442380/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(12/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Table[31^n*HermiteH[n, 6/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 6/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 6/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(12*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 12*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 6/31).
E.g.f.: exp(12*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(12/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160305 Numerator of Hermite(n, 7/31).

Original entry on oeis.org

1, 14, -1726, -77980, 8860396, 723555784, -75018624584, -9394306045264, 877780290519440, 156735773819251424, -12989542631935753184, -3194315169653112913856, 229904497949242113022144, 76892348044168785827484800, -4667900913141400434386502784
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 14/31, -1726/961, -77980/29791, 8860396/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(14/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Numerator[HermiteH[Range[0, 20], 7/31]] (* Harvey P. Dale, Apr 23 2016 *)
Table[31^n*HermiteH[n, 7/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 7/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 7/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(14*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 14*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 7/31).
E.g.f.: exp(14*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(14/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160306 Numerator of Hermite(n, 8/31).

Original entry on oeis.org

1, 16, -1666, -88160, 8195596, 808903616, -65817219704, -10381352014976, 719403241658000, 171134120448798976, -9706091347019300384, -3444495256578225124864, 150094259153430446720704, 81845346744175071427394560, -2440729611300811998925197184
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2009

Keywords

Examples

			Numerators of 1, 16/31, -1666/961, -88160/29791, 8195596/923521, ...

Links

Crossrefs

Cf. A009975 (denominators).

Programs

  • Magma
    [Numerator((&+[(-1)^k*Factorial(n)*(16/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018
  • Mathematica
    Table[31^n*HermiteH[n, 8/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)
  • Maxima
    makelist(num(hermite(n, 8/31)), n, 0, 20); /* Bruno Berselli, Mar 28 2018 */
  • PARI
    a(n)=numerator(polhermite(n, 8/31)) \\ Charles R Greathouse IV, Jan 29 2016
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(16*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

Formula

a(n+2) = 16*a(n+1) - 1922*(n+1)*a(n). - Bruno Berselli, Mar 28 2018
From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 8/31).
E.g.f.: exp(16*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(16/31)^(n-2*k)/(k!*(n-2*k)!)). (End)
Previous Showing 11-20 of 44 results. Next

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

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