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 1-3 of 3 results.

A285270 a(n) = H_n(n), where H_n is the physicist's n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 14, 180, 3340, 80600, 2389704, 83965616, 3409634960, 157077960480, 8093278209760, 461113571640128, 28784033772836544, 1953535902100115840, 143219579014652040320, 11279408109860685024000, 949705205977314865582336, 85131076752851318807814656, 8094279370190580822082014720
Offset: 0

Views

Author

Natan Arie Consigli, May 24 2017

Keywords

Examples

			Knowing that H_3(x) = 8x^3-12x, a(3) = H_3(3) = 8*3^3-12*3 = 180.

Links

Crossrefs

Cf. A089466 (probabilist's variant).
Cf. A349066, A349067, A349068, A349069.

Programs

  • Mathematica
    Table[HermiteH[n, n], {n, 0, 18}] (* Michael De Vlieger, May 25 2017 *)
  • PARI
    a(n) = polhermite(n, n); \\ Michel Marcus, May 25 2017
  • Python
    from sympy import hermite
def a(n): return hermite(n, n) # Indranil Ghosh, May 25 2017

Formula

a(n) ~ exp(-1/4) * 2^n * n^n. - Vaclav Kotesovec, Nov 07 2021

Extensions

More terms from Michel Marcus, May 25 2017

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
Showing 1-3 of 3 results.

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

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