Theodore Kolokolnikov - cp's OEIS frontend

User: Theodore Kolokolnikov

Theodore Kolokolnikov has authored 2 sequences.

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 3, 4, 6, 5, 6, 7, 8, 8, 10, 11, 16, 16, 19, 21, 28, 29, 34, 41, 50, 56, 68, 80, 100, 114, 135, 158, 196, 225, 269, 320, 388, 455, 544, 644, 786, 921, 1111, 1321, 1600, 1891, 2274, 2711, 3280, 3895, 4694, 5591, 6780, 8051, 9729, 11624

Offset: 0

Theodore Kolokolnikov, May 01 2009

nonn

If n is even then a(n) is the absolute value of the coefficient of z^(n(n+1)/4). If n is odd, it is an open question as to which coefficient is a(n).

For odd n values, the Berkovich/Uncu reference provides explicit conjectural formulas for a(n). - Ali Uncu, Jul 19 2020

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms n = 1..100 from Theodore Kolokolnikov, terms n = 101..1000 from Alois P. Heinz)
Alexander Berkovich, and Ali K. Uncu, Where do the maximum absolute q-series coefficients of (1-q)(1-q^2)(1-q^3)...(1-q^(n-1))(1-q^n) occur?, arXiv:1911.03707 [math.NT], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A063866, A069918, A133871.

A160089 := proc(n)
        g := expand(mul( 1-x^k,k=1..n) );
        convert(PolynomialTools[CoefficientVector](g, x), list):
        max(op(map(abs, %)));
end proc:

p = 1; Flatten[{1, Table[p = Expand[p*(1 - x^n)]; Max[Abs[CoefficientList[p, x]]], {n, 1, 100}]}] (* Vaclav Kotesovec, May 03 2018 *)

a(n) >= A086376(n). - R. J. Mathar, Jun 01 2011
From Vaclav Kotesovec, May 04 2018: (Start)
a(n)^(1/n) tends to 1.2197...
Conjecture: a(n)^(1/n) ~ sqrt(A133871(n)^(1/n)) ~ 1.21971547612163368901359933...
(End)

a(0)=1 prepended by Alois P. Heinz, Apr 12 2017

A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 14, 21, 32, 54, 87, 144, 230, 383, 671, 1158, 1981, 3408, 6246, 10925, 19463, 34624, 63941, 114954, 208429, 380130, 707194, 1298600, 2379842, 4398644, 8253618, 15303057, 28453809, 53091455, 100061278, 187446097

Offset: 0

Theodore Kolokolnikov, May 05 2009

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..200 from Seiichi Manyama)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A033461, A359319, A359320, A369728.

for N from 1 to 40 do
p := expand(product(1+x^(n^2), n=1..N)):
L:=convert(PolynomialTools[CoefficientVector](p, x), list):
mmax := max(op(map(abs, L)));
lprint(mmax):
end:

p = 1; Table[p = Expand[p*(1 + x^(n^2))]; Max[CoefficientList[p, x]], {n, 1, 50}] (* Vaclav Kotesovec, May 04 2018 *)
nmax = 100; poly = ConstantArray[0, nmax*(nmax+1)*(2*nmax+1)/6 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, k*(k+1)*(2*k+1)/6, k^2, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 30 2022 *)

An asymptotic formula is a(n) ~ sqrt(10/Pi) * n^(-5/2) * 2^n. See for example the reference by Finch.

More precise asymptotics: a(n) ~ sqrt(10/Pi) * 2^n / n^(5/2) * (1 - 35/(18*n) + ...). - Vaclav Kotesovec, Dec 30 2022

a(0)=1 prepended by Seiichi Manyama, Dec 26 2022

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User: Theodore Kolokolnikov

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

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A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).

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cp's OEIS Frontend

User: Theodore Kolokolnikov

A160089 The maximum of the absolute value of the coefficients of Pn = (1-x)(1-x^2)(1-x^3)...(1-x^n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A160235 The maximal coefficient of (1+x)*(1+x^4)*(1+x^9)*...*(1+x^(n^2)).

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A160235 The maximal coefficient of (1+x)(1+x^4)(1+x^9)...(1+x^(n^2)).