A381524 Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.
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
Keywords
Examples
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.
Programs
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Mathematica
Table[(PositionLargest@Abs[CoefficientList[HermiteH[n, x], x]])[[1]] - 1, {n, 0, 100}] -
PARI
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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