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.

A277281 Maximal coefficient (ignoring signs) in Hermite polynomial of order n.

Original entry on oeis.org

1, 2, 4, 12, 48, 160, 720, 3360, 13440, 80640, 403200, 2217600, 13305600, 69189120, 484323840, 2905943040, 19372953600, 131736084480, 846874828800, 6436248698880, 42908324659200, 337903056691200, 2477955749068800, 18997660742860800, 151981285942886400
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 08 2016

Keywords

Examples

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient (ignoring signs) is 160, so a(5) = 160.

Links

Crossrefs

Cf. A059343, A277280 (with signs).

Programs

  • Mathematica
    Table[Max@Abs@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]
  • PARI
    a(n) = vecmax(apply(x->abs(x), Vec(polhermite(n)))); \\ Michel Marcus, Oct 09 2016
  • Python
    from sympy import hermite, Poly
def a(n): return max(map(abs, Poly(hermite(n, x), x).coeffs())) # Indranil Ghosh, May 26 2017

A277378 Expansion of e.g.f. exp(2*x/(1-x))/sqrt(1-x^2).

Original entry on oeis.org

1, 2, 9, 50, 361, 3042, 29929, 331298, 4100625, 55777922, 828691369, 13316140818, 230256982201, 4257449540450, 83834039024649, 1750225301567618, 38614608429012001, 897325298084953602, 21904718673762721225, 560258287738117292018, 14981472258320814527241
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 11 2016

Keywords

Links

Crossrefs

Cf. A000321, A000898, A059343, A062267, A067994, A277280, A277281.

Programs

  • Mathematica
    Table[Abs[HermiteH[n, I]]^2/2^n, {n, 0, 20}]
With[{nn=20},CoefficientList[Series[Exp[2x/(1-x)]/Sqrt[1-x^2],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 27 2023 *)

Formula

E.g.f.: exp(2*x/(1-x))/sqrt(1-x^2).
a(n) = |H_n(i)|^2 / 2^n = H_n(i) * H_n(-i) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+2)*(a(n) + n*a(n-1)) = a(n+1) + n*(n-1)^2*a(n-2).
a(n) ~ n^n / (2 * exp(1 - 2*sqrt(2*n) + n)) * (1 + 2*sqrt(2)/(3*sqrt(n))). - Vaclav Kotesovec, Oct 27 2021

A277379 E.g.f.: exp(x/(1-x^2))/sqrt(1-x^2).

Original entry on oeis.org

1, 1, 2, 10, 40, 296, 1936, 17872, 164480, 1820800, 21442816, 279255296, 3967316992, 59837670400, 988024924160, 17009993230336, 318566665977856, 6177885274406912, 129053377688043520, 2786107670662021120, 64136976817284448256, 1525720008470138454016
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 11 2016

Keywords

Comments

Is this the same as A227545 (at least for n>=1)?

Links

Crossrefs

Cf. A000321, A000898, A059343, A062267, A067994, A227545, A277280, A277281, A277378.

Programs

  • Mathematica
    Table[Abs[HermiteH[n, (1 + I)/2]]^2/2^n, {n, 0, 20}]

Formula

a(n) = |H_n((1+i)/2)|^2 / 2^n = H_n((1+i)/2) * H_n((1-i)/2) / 2^n, where H_n(x) is n-th Hermite polynomial, i = sqrt(-1).
D-finite with recurrence: (n+1)*(n+2)*(a(n) - n^2*a(n-1)) + (2*n^2+7*n+6)*a(n+1) + a(n+2) = a(n+3).
a(n) ~ n^n * exp(sqrt(2*n)-n) / 2. - Vaclav Kotesovec, Oct 14 2016

A381423 Exponent of x of maximal coefficient in Hermite polynomial of order n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 7, 8, 9, 10, 11, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 9, 10, 11, 12, 9, 10, 11, 12, 13, 10
Offset: 0

Views

Author

Mike Sheppard, Feb 23 2025

Keywords

Comments

The exponent is always unique. The coefficients, in absolute value, follow a unimodal pattern, and their signs alternate. If the maximum absolute coefficient appears twice due to symmetry (e.g., H_8(x)), the terms will have opposite signs, ensuring a unique exponent for the maximum signed coefficient.
Conjecture: Differences are either 1 or -3; more specifically the patterns (1,1,1,-3) or (1,1,1,1,-3), with position of those patterns appearing at linearly and quadratically spaced intervals, respectively. Seems to grow O(n^(1/2))

Examples

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient is 120 (we take signs into account, so -160 < 120), occurring at x^1, hence a(5) = 1.

Crossrefs

Cf. A277280 (maximal coefficient).

Programs

  • Mathematica
    Table[(PositionLargest@CoefficientList[HermiteH[n, x], x])[[1]] - 1, {n, 0, 100}]
  • PARI
    a(n) = my(p=polhermite(n), m=vecmax(Vec(p))); for(i=0, poldegree(p), if (polcoef(p, i) == m, return(i))); \\ Michel Marcus, Feb 23 2025
Showing 1-4 of 4 results.

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