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 1-4 of 4 results.

A349066 a(n) = H(2*n, n), where H(n,x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 76, 14136, 5324432, 3275529760, 2982971060928, 3773262142004096, 6332628384952750336, 13620318069121988018688, 36536710970888029776972800, 119598502032157660592768038912, 469232422933986002753883881312256, 2173747962477936168042899607178059776
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A285270, A349067, A349068.

Programs

  • Maple
    a:= n-> simplify(HermiteH(2*n, n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 07 2021
  • Mathematica
    Table[HermiteH[2*n, n], {n, 0, 15}]
  • PARI
    a(n) =  polhermite(2*n, n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) ~ exp(-1) * 2^(2*n) * n^(2*n).

A349067 a(n) = H(3*n, n), where H(n,x) is n-th Hermite polynomial.

Original entry on oeis.org

1, -4, -824, -406944, 854857408, 36727035808000, 1350597603460566528, 70169228831160001808384, 5261285254051930823802720256, 556216363355718012207356567863296, 80574670961706857240366003306352640000, 15573012689517863187913236259514917169004544
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

In general, for k>=1, H(k*n,n) ~ exp(-k^2/4) * 2^(k*n) * n^(k*n).

Links

Crossrefs

Cf. A285270, A349066, A349069.

Programs

  • Maple
    a:= n-> simplify(HermiteH(3*n, n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 07 2021
  • Mathematica
    Table[HermiteH[3*n, n], {n, 0, 12}]
  • PARI
    a(n) =  polhermite(3*n, n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) ~ exp(-9/4) * 2^(3*n) * n^(3*n).

A349068 a(n) = H(n, 2*n), where H(n,x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 4, 62, 1656, 62476, 3041200, 181253256, 12779289376, 1040259450512, 96008691963456, 9906193528929760, 1129945699713533824, 141183268107518731968, 19176614030629200880384, 2813353012562289110458496, 443345766248682440278848000, 74687922008799389150557901056
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A285270, A349066, A349069.

Programs

  • Maple
    a:= n-> simplify(HermiteH(n, 2*n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 07 2021
  • Mathematica
    Table[HermiteH[n, 2*n], {n, 0, 20}]
  • PARI
    a(n) =  polhermite(n, 2*n); \\ Michel Marcus, Nov 07 2021

Formula

a(n) ~ exp(-1/16) * 4^n * n^n.
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * ( n! / (k! * (n-2k)!) ) * (4n)^(n-2k), for n>0. - Bernard Schott, Nov 07 2021

A349069 a(n) = H(n, 3*n), where H(n,x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 6, 142, 5724, 324876, 23761800, 2126627016, 225081383184, 27498818692752, 3808595968290144, 589662462800129760, 100917872425324633536, 18918488805502510634688, 3855242696428245589623936, 848531650317994634533024896, 200604383862593153678170272000
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Comments

In general, for k>=1, H(n,k*n) ~ exp(-1/(4*k^2)) * (2*k)^n * n^n.

Links

Crossrefs

Cf. A285270, A349067, A349068.

Programs

  • Maple
    a:= n-> simplify(HermiteH(n, 3*n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 07 2021
  • Mathematica
    Table[HermiteH[n, 3*n], {n, 0, 20}]

Formula

a(n) ~ exp(-1/36) * 6^n * n^n.
Showing 1-4 of 4 results.

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