A058377 -id:A058377 - cp's OEIS frontend

A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

1, 0, 1, 0, 1, 0, 1, 1, 4, 5, 9, 15, 26, 45, 77, 137, 243, 434, 774, 1408, 2554, 4667, 8627, 15927, 29559, 54867, 101688, 189425, 355315, 668598, 1264180, 2395462, 4506221, 8507311, 16084405, 30545142, 57898862, 110199367, 209957460, 400430494, 765333684

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000040, A022894, A058377, A063890, A113040, A261061, A350404.

Table[SeriesCoefficient[Product[x^Prime[k] + 1/x^Prime[k], {k, n - 1}], {x, 0, n}], {n, 0, 40}] (* Stefano Spezia, Jan 30 2022 *)

from sympy import sieve, primerange
from functools import cache
@cache
def b(n, i):
    maxsum = 0 if i < 2 else sum(p for p in primerange(2, sieve[i-1]+1))
    if n > maxsum: return 0
    if i < 2: return 1
    return b(n+sieve[i-1], i-1) + b(abs(n-sieve[i-1]), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(41)]) # Michael S. Branicky, Jan 29 2022

a(n) = [x^n] Product_{k=1..n-1} (x^prime(k) + 1/x^prime(k)).

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

Original entry on oeis.org

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

A113035 Number of ways the set {1,2,...,n} can be split into two subsets of which the sum of one is twice the sum of the other.

Original entry on oeis.org

0, 1, 1, 0, 3, 4, 0, 10, 17, 0, 46, 78, 0, 231, 401, 0, 1233, 2177, 0, 6869, 12268, 0, 39502, 71172, 0, 232686, 422076, 0, 1396669, 2547246, 0, 8512170, 15593760, 0, 52534875, 96598865, 0, 327669853, 604405633, 0, 2062171364, 3814087419, 0, 13078921499

Offset: 1

Floor van Lamoen, Oct 11 2005

nonn

			For n=5 we have 5/1234, 14/532 and 23/541 so a(5)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A058377, A112972.

A113035:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-2*i)+x^(i)); od; t:=t,coeff(p,x,0); od; t; end;
# second Maple program:
b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
    end:
a:= n-> `if`(irem(n, 3)=1, 0, b(n*(n+1)/6, n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 31 2011

b[n_, i_] := b[n, i] = Module[{m = i(i+1)/2}, If[n > m, 0, If[n == m, 1, b[Abs[n - i], i - 1] + b[n + i, i - 1]]]];
a[n_] := If[Mod[n, 3] == 1, 0, b[n(n+1)/6, n]];
Array[a, 60] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

a(n) is the coefficient of x^0 in Product_{k=1..n} x^(-2k)+x^k.

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A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

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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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A113035 Number of ways the set {1,2,...,n} can be split into two subsets of which the sum of one is twice the sum of the other.

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Maple

Mathematica

Formula