A277281 -id:A277281 - cp's OEIS frontend

A277280 Maximal coefficient in Hermite polynomial of order n.

1, 2, 4, 8, 16, 120, 720, 3360, 13440, 48384, 302400, 2217600, 13305600, 69189120, 322882560, 2421619200, 19372953600, 131736084480, 790416506880, 4290832465920, 40226554368000, 337903056691200, 2477955749068800, 16283709208166400, 113985964457164800

Offset: 0

Vladimir Reshetnikov, Oct 08 2016

nonn

			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), hence a(5) = 120.

Vaclav Kotesovec, Table of n, a(n) for n = 0..710
Vaclav Kotesovec, Plot of a(n+1)/(a(n)*sqrt(n)) for n = 1..10000
Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

Cf. A059343, A277281 (ignoring signs).

Table[Max@CoefficientList[HermiteH[n, x], x], {n, 0, 25}]

a(n) = vecmax(Vec(polhermite(n))); \\ Michel Marcus, Oct 09 2016

from sympy import hermite, Poly
def a(n): return max(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

Vladimir Reshetnikov, Oct 11 2016

nonn

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

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

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 *)

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

A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

Original entry on oeis.org

8, 13, 34, 43, 76, 89, 134, 151, 208, 229, 298, 323, 404, 433, 526, 559, 664, 701, 818, 859, 988, 1033, 1174, 1223, 1376, 1429, 1594, 1651, 1828, 1889, 2078, 2143, 2344, 2413, 2626, 2699, 2924, 3001, 3238, 3319, 3568, 3653, 3914, 4003, 4276, 4369, 4654, 4751, 5048

Offset: 1

Mike Sheppard, Mar 05 2025

nonn

			H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8, maximum coefficient in absolute value is 13440, which appears twice. Hence 8 is a term.
H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6. Absolute maximum unique. Hence 6 is not a term.

Mike Sheppard, Table of n, a(n) for n = 1..70

Cf. A060821, A277281, A381524.

Flatten@Position[Table[Count[#, Max@#] &@Abs@CoefficientList[HermiteH[n, x], x], {n, 1000}], 2]

isok(k) = my(vp=apply(x->abs(x), Vec(polhermite(k))), m=vecmax(vp)); #select(x->(x==m), vp) == 2; \\ Michel Marcus, Mar 09 2025

Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).
Conjecture 2: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
Conjecture 3: G.f.: (8*x + 5*x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).
Terms < 20000 consistent with conjectures. - Jinyuan Wang, Mar 09 2025.

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

Vladimir Reshetnikov, Oct 11 2016

nonn

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

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Wikipedia, Hermite polynomials.

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

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

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

A381524 Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.

Original entry on oeis.org

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

Offset: 0

Mike Sheppard, Feb 26 2025

nonn

Exponent is unique except for order of n within A381737, whose maximum of absolute value of coefficients appear twice. For example, H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8, maximum coefficient in absolute value is 13440, which appears twice. For those values a(n) and a(n)+2 both are maximums, in absolute value.

Conjecture: Differences are either +1 or -1.

			For n = 5, H_5(x) = 32*x^5 - 160*x^3 + 120*x. The maximal coefficient (ignoring signs) is 160, occurring at x^3, hence a(5) = 3.
For n = 8, H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8. The maximal coefficient (ignoring signs) is 13440, occurring at both x^2 and x^4, the smallest exponent being 2, hence a(8) = 2.

Cf. A277281 (maximal coefficient ignoring signs), A381737 (non-unique exponents).

Table[(PositionLargest@Abs[CoefficientList[HermiteH[n, x], x]])[[1]] - 1, {n, 0, 100}]

a(n) = my(p=polhermite(n), m=vecmax(apply(x->abs(x), Vec(p)))); for(i=0, poldegree(p), if (abs(polcoef(p, i)) == m, return(i))); \\ Michel Marcus, Feb 26 2025

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A277280 Maximal coefficient in Hermite polynomial of order n.

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

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A381737 Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.

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

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A381524 Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.

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Comments

Examples

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Programs

Mathematica

PARI