A158380 -id:A158380 - cp's OEIS frontend

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0, 29058, 45106, 0, 0, 276828, 473854, 0, 0, 2455340, 4777436, 0, 0, 27466324, 46429640, 0, 0, 280395282, 526489336, 0, 0, 3193589950, 5661226928, 0, 0

Offset: 1

Pietro Majer, Mar 19 2009

nonn

Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).
a(n)=0 for any n=1 (mod 4) or n=2 (mod 4).
Andrica & Tomescu give a more general integral formula than the one below. The asymptotic formula below is a conjecture by Andrica & Ionascu; it remains unproven. - Jonathan Sondow, Nov 11 2013

			For n=16 the a(16) = 2 solutions are +1 +16 +81 +256 -625 -1296 -2401 +4096 +6561 +10000 +14641 +20736 -28561 -38416 -50625 +65536 = 0 and the opposite.

D. Andrica and E. J. Ionascu, Variations on a result of Erdős and SurányiINTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Cf. A063865, A158092, A158118, A158380, A019568.

A111253(n) = a(n)/2. - Alois P. Heinz, Oct 31 2011

N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p,x,0)]: od:a;

Integral representation: a(n) = ((2^n)/Pi)*int_0^pi prod_{k=1}^n cos(x*k^4) dx.

Asymptotic formula: a(n) = (2^n)*sqrt(18/(Pi*n^9))*(1+o(1)) as n->infinity; n=-1 or 0 (mod 4).

a(35)-a(58) from Alois P. Heinz, Oct 31 2011

A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n(3n-1)/2 = 0.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 4, 4, 0, 30, 46, 78, 0, 210, 366, 644, 0, 2032, 3696, 6694, 0, 21936, 39886, 73098, 0, 246172, 454074, 841714, 0, 2899542, 5401222, 10073398, 0, 35282910, 66213604, 124427582, 0, 441326270, 832775792, 1573861942, 0, 5642205488

Offset: 0

Seiichi Manyama, Sep 17 2017

nonn

			For n=6 the 2 solutions are +1+5-12+22+35-51 = 0 and -1-5+12-22-35+51 = 0.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A000326, A002411, A158092, A158380, A292475.

{a(n) = polcoeff(prod(k=1, n, x^(k*(3*k-1)/2)+1/x^(k*(3*k-1)/2)), 0)}

Constant term in the expansion of Product_{k=1..n} (x^(k*(3*k-1)/2)+1/x^(k*(3*k-1)/2)).

a(4*k+1) = 0 for k >= 0.

A359348 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 12, 18, 27, 44, 73, 122, 210, 362, 620, 1050, 1857, 3290, 5949, 10665, 19086, 34330, 62252, 113643, 209460, 383888, 706457, 1300198, 2407535, 4468367, 8331820, 15525814, 28987902, 54180854, 101560631, 190708871, 358969426

Offset: 0

Seiichi Manyama, Dec 27 2022

nonn

			(1 + x) * (1 + x^3) * (1 + x^6) * (1 + x^10) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2 * x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + x^19 + x^20. So a(4) = 2.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A000217, A024940, A025591, A158380, A160235.

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k*(k+1)/2))));

a(n) ~ sqrt(5) * 2^(n + 3/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Dec 29 2022

A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 11, 27, 61, 133, 311, 761, 1839, 4575, 11573, 29641, 76487, 199617, 524067, 1384697, 3681069, 9841217, 26437741, 71369101, 193496241, 526685793, 1438816755, 3944034221, 10845006963, 29908325821, 82707648985, 229306378067, 637283978821

Offset: 0

Ilya Gutkovskiy, Jan 20 2024

nonn

All terms are odd.

a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * i*(i+1)/2 with c_i in {-1,0,1}.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000217, A007576, A158380, A350249.

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(i+2)/6, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+i*(i+1)/2, i-1)+b(abs(n-i*(i+1)/2), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024

Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, 0], {n, 0, 31}]

a(n) ~ sqrt(5) * 3^(n + 1/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jan 22 2024

A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 3, 0, 4, 3, 9, 0, 27, 43, 71, 0, 190, 318, 604, 0, 1846, 3127, 5664, 0, 19048, 34065, 62045, 0, 205713, 378243, 705836, 0, 2403370, 4434940, 8276125, 0, 28980680, 54167797, 101541048, 0, 358095372, 674776903, 1274888645, 0, 4551828850, 8612421500

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000217, A058498, A063890, A158380, A348165, A350287.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(i+2)//6: return 0
    if i == 0: return 1
    return b(n+i*(i+1)//2, i-1) + b(abs(n-i*(i+1)//2), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 29 2022

a(n) = [x^n] Product_{k=1..n} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)).

A292510 a(n) = smallest k >= 1 such that {1, p(n,2), p(n,3), ..., p(n,k)} can be partitioned into two sets with equal sums, where p(n,m) is m-th n-gonal number.

Original entry on oeis.org

4, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 3

Seiichi Manyama, Sep 17 2017

nonn

Conjecture: a(n) = 7 for n > 5.

			n = 3
1+3+6 = 10
n = 4
1+4+16+49 = 9+25+36 (= 70 = 28*4-42)
n = 5
1+5+22+35 = 12+51 (=63)
n = 6
1+6+28+91 = 15+45+66 (= 126 = 28*6-42)

Wikipedia, Polygonal number

Cf. A019568, A158092, A158380, A292474.

def f(k, n)
  n * ((k - 2) * n - k + 4) / 2
end
def A(k, n)
  ary = [1]
  s_ary = [0]
  (1..n).each{|i| s_ary << s_ary[-1] + f(k, i)}
  m = s_ary[-1]
  a = Array.new(m + 1){0}
  a[0] = 1
  (1..n).each{|i|
    b = a.clone
    (0..[s_ary[i - 1], m - f(k, i)].min).each{|j| b[j + f(k, i)] += a[j]}
    a = b
    s_ary[i] % 2 == 0 ? ary << a[s_ary[i] / 2] : ary << 0
  }
  ary
end
def B(n)
  i = 1
  while A(n, i)[-1] == 0
    i += 1
  end
  i
end
def A292510(n)
  (3..n).map{|i| B(i)}
end
p A292510(100)

p(n,1) + p(n,2) + p(n,4) + p(n,7) = p(n,3) + p(n,5) + p(n,6) (= 28*n-42). So a(n) <= 7.

A350287 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = 0 or 1.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 2, 4, 7, 12, 16, 26, 42, 66, 104, 210, 318, 620, 970, 1748, 3281, 5948, 10480, 18976, 34233, 60836, 111430, 209460, 378529, 704934, 1284836, 2387758, 4466874, 8331820, 15525814, 28987902, 54162165, 101242982, 190267598, 358969426, 674845081

Offset: 0

Ilya Gutkovskiy, Jan 19 2022

nonn

			a(4) = 2: -1 - 3 - 6 + 10 = +1 + 3 + 6 - 10 = 0.

Cf. A000217, A025591, A058498, A158380.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(i+2)//6: return 0
    if i == 0: return 1
    return b(n+i*(i+1)//2, i-1) + b(abs(n-i*(i+1)//2), i-1)
def a(n): return b(0, n) + b(1, n)
print([a(n) for n in range(41)]) # Michael S. Branicky, Jan 19 2022

A369494 a(n) = [x^(n(n+1)/2)] Product_{k=1..n} (x^(k(k+1)/2) + 1/x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 3, 3, 5, 0, 14, 23, 39, 0, 101, 161, 315, 0, 971, 1595, 2872, 0, 9697, 17431, 31736, 0, 103608, 190242, 356883, 0, 1218049, 2235343, 4165201, 0, 14602056, 27304610, 51182196, 0, 179995388, 339041695, 640927871, 0, 2288387318, 4326722468, 8201714149

Offset: 0

Ilya Gutkovskiy, Jan 24 2024

nonn

Cf. A000217, A063890, A158380, A351002, A368243.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
     `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+
        b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)=2, 0, b(n*(n+1)/2, n)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 24 2024

Table[Coefficient[Product[x^(k (k + 1)/2) + 1/x^(k (k + 1)/2), {k, 1, n}], x, n (n + 1)/2], {n, 0, 45}]

cp's OEIS Frontend

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

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A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n*(3*n-1)/2 = 0.

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A359348 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

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A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

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A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

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A292510 a(n) = smallest k >= 1 such that {1, p(n,2), p(n,3), ..., p(n,k)} can be partitioned into two sets with equal sums, where p(n,m) is m-th n-gonal number.

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A350287 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = 0 or 1.

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Python

A369494 a(n) = [x^(n*(n+1)/2)] Product_{k=1..n} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)).

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Maple

Mathematica

A292474 Number of solutions to +-1 +- 5 +- 12 +- ... +- n(3n-1)/2 = 0.

A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

A369494 a(n) = [x^(n(n+1)/2)] Product_{k=1..n} (x^(k(k+1)/2) + 1/x^(k*(k+1)/2)).