A000980 -id:A000980 - cp's OEIS frontend

A379451 Number of ordered ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

2, 10, 162, 6278, 430906, 46032666, 7029940154, 1453778429782, 390651831405906, 132345369222827306, 55150093300481888770, 27727437337790844360198, 16545310955942988999292586, 11561068480810074519638819626, 9349537740123803513263001013354, 8664632430514446774520557369434870

Offset: 0

Ilya Gutkovskiy, Dec 23 2024

nonn

			a(1) = 10 ways: {}, {0}, {-1, 1} (2 permutations), {-1, 0, 1} (6 permutations).

Michael S. Branicky, Table of n, a(n) for n = 0..100

Cf. A000980, A058377, A063865.

from math import factorial
from functools import cache
@cache
def b(i, s, c):
    if i == 0: return factorial(c) if s == 0 else 0
    return b(i-1, s, c) + b(i-1, s+(i>>1)*(-1)**(i&1), c+1)
def a(n): return b(2*n+1, 0, 0)
print([a(n) for n in range(16)]) # Michael S. Branicky, Dec 23 2024

a(9) and beyond from Michael S. Branicky, Dec 23 2024

A156082 Maximum coefficient of the polynomial (-1)^(n+1)*Product_{k=1..n} (1 - x^k)^2.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 19, 24, 36, 52, 74, 103, 156, 223, 322, 470, 682, 992, 1448, 2120, 3072, 4494, 6538, 9584, 14001, 20400, 29928, 43774, 64032, 93968, 137520, 201766, 296236, 433746, 637812, 936334, 1373622, 2021344, 2968872, 4364300, 6422472

Offset: 1

Steven Finch, Feb 03 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..2405
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A133871.

P:= -1:
for n from 1 to 100 do
  P:= expand(-P*(1-x^n)^2);
  A[n]:= max(coeffs(P,x));
od:
seq(A[i],i=1..100); # Robert Israel, Mar 02 2018

Table[ -Min[CoefficientList[Expand[(-1)^n*Product[(1 - x^k)^2, {k, 1, n}]],x]], {n, 1, 50}]

Name edited by Robert Israel, Mar 02 2018

cp's OEIS Frontend

A379451 Number of ordered ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

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