A025591 -id:A025591 - cp's OEIS frontend

A083527 a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns.

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 5, 0, 0, 43, 57, 0, 0, 239, 430, 0, 0, 2904, 5419, 0, 0, 27813, 50213, 0, 0, 348082, 649300, 0, 0, 3913496, 7287183, 0, 0, 50030553, 93696497, 0, 0, 611793542, 1161079907, 0, 0, 8009933135, 15176652567, 0, 0

Offset: 1

T. D. Noe, Apr 29 2003

nonn

The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums.
a(n) is the maximal number of subsets of the first n squares that share the same sum. Cf. A025591, A083309.
a(n)=0 when n==1 or 2 (mod 4).

			a(7) = 1 because there is only one sign pattern of the first seven squares that yields zero: 1+4-9+16-25-36+49.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
T. D. Noe, Extremal Sums of Sequences

Cf. A015818, A058498, A063865, A113263, A158092.

b:= proc(n, i) option remember; local m;
      m:= (1+(3+2*i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 31 2011

d={1, 1}; nMax=60; zeroLst={0}; Do[p=n^2; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[1==Mod[Length[d], 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]], AppendTo[zeroLst, 0]], {n, 2, nMax}]; zeroLst/2
p = 1; t = {}; Do[p = Expand[p(x^(n^2) + x^(-n^2))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 51}]; t (* Robert G. Wilson v, Oct 31 2005 *)

a(n)=sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012

a(n) is half the coefficient of x^0 in the product_{k=1..n} x^(k^2)+x^(k^-2).
a(n) = A158092(n)/2.
a(n) = [x^(n^2)] Product_{k=1..n-1} (x^(k^2) + 1/x^(k^2)). - Ilya Gutkovskiy, Feb 01 2024

A350457 Maximal coefficient of (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 7, 10, 16, 27, 45, 79, 139, 249, 439, 784, 1419, 2574, 4703, 8682, 16021, 29720, 55146, 102170, 190274, 356804, 671224, 1269022, 2404289, 4521836, 8535117, 16134474, 30635869, 58062404, 110496946, 210500898, 401422210, 767158570, 1467402238

Offset: 0

Ilya Gutkovskiy, Jan 01 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A000040, A000586, A007504, A025591, A160235.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1+x^ithprime(n))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 01 2022

b[n_] := b[n] = If[n == 0, 1, Expand[(1 + x^Prime[n])*b[n - 1]]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

a(n) = vecmax(Vec(prod(k=1, n, 1 + x^prime(k)))); \\ Michel Marcus, Jan 01 2022

from sympy.abc import x
from sympy import prime, prod
def A350457(n): return 1 if n == 0 else max(prod(1+x**prime(i) for i in range(1,n+1)).as_poly().coeffs()) # Chai Wah Wu, Jan 03 2022

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

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

Original entry on oeis.org

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

A063867 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or +- 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 10, 8, 14, 46, 80, 70, 124, 442, 794, 722, 1314, 4820, 8882, 8220, 15272, 56920, 106444, 99820, 187692, 707486, 1336546, 1265204, 2399784, 9119656, 17358560, 16547220, 31592878, 120801376, 231266520, 221653776

Offset: 0

N. J. A. Sloane, following a suggestion by J. H. Conway, Aug 27 2001

Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000; first 101 terms from T. D. Noe)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A063865, A063866.

f[n_, s_] := f[n, s]=Which[n==0, If[s==0, 1, 0], Abs[s]>(n*(n+1))/2, 0, True, f[ n-1, s-n]+f[n-1, s+n]]; a[n_] := f[n, 0]+2f[n, 1]

a(n) = A063865(n) + 2*A063866(n).

More terms from Dean Hickerson and Vladeta Jovovic, Aug 28 2001

A039828 Largest coefficient in expansion of Product_{i=1..n} (1 + (-q)^i).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 23, 40, 69, 123, 221, 397, 722, 1314, 2410, 4441, 8215, 15260, 28460, 53222, 99820, 187692, 353743, 668273, 1264854, 2399207, 4559828, 8679280, 16547220, 31592878, 60400688, 115633260, 221625505, 425313967

Offset: 1

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A025591, A039829.

a(n) = vecmax(Vec(prod(i = 1, n, (1 + (-q)^i)))); \\ Michel Marcus, Aug 30 2018

a(n) ~ sqrt(3) * 2^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 04 2022

A069918 Number of ways of partitioning the set {1...n} into two subsets whose sums are as nearly equal as possible.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 4, 7, 23, 40, 35, 62, 221, 397, 361, 657, 2410, 4441, 4110, 7636, 28460, 53222, 49910, 93846, 353743, 668273, 632602, 1199892, 4559828, 8679280, 8273610, 15796439, 60400688, 115633260, 110826888, 212681976, 817175698, 1571588177, 1512776590

Offset: 1

Robert G. Wilson v, Apr 24 2002

nonn

If n mod 4 = 0 or 3, a(n) is the number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1; if n mod 4 = 1 or 2, a(n) is half this number.

			If the triangular number T_n (see A000217) is even then the two totals must be equal, otherwise the two totals differ by one.
a(6) = 5: T6 = 21 and is odd. There are five sets such that the sum of one side is equal to the other side +/- 1. They are 5+6 = 1+2+3+4, 4+6 = 1+2+3+5, 1+4+6 = 2+3+5, 1+3+6 = 2+4+5 and 2+3+6 = 1+4+5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
S. R. Finch, Signum equations and extremal coefficients [broken link]
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Brian Hayes, The Easiest Hard Problem, American Scientist, Vol. 90, No. 2, March-April 2002, pp. 113-117.
Recreational Mathematics Newsletter, Kolstad, PROBLEM 4: Subset Sums [broken link]
Dave Rusin, More Number Theory [broken link]

Cf. A060005, A058377, A025591, A063865, A063866, A063867, A306443.

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-1, 4)<2, b(n-1, n-1) +b(n+1, n-1), b(n, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 02 2011

Needs["DiscreteMath`Combinatorica`"]; f[n_] := f[n] = Block[{s = Sort[Plus @@@ Subsets[n]], k = n(n + 1)/2}, If[ EvenQ[k], Count[s, k/2]/2, (Count[s, Floor[k/2]] + Count[s, Ceiling[k/2]]) /2]]; Table[ f[n], {n, 1, 22}]
f[n_, s_] := f[n, s] = Which[n == 0, If[s == 0, 1, 0], Abs[s] > (n*(n + 1))/2, 0, True, f[n - 1, s - n] + f[n - 1, s + n]]; Table[ Which[ Mod[n, 4] == 0 || Mod[n, 4] == 3, f[n, 0]/2, Mod[n, 4] == 1 || Mod[n, 4] == 2, f[n, 1]], {n, 1, 40}]

If n mod 4 = 0 or 3 then the two subsets have the same sum and a(n) = A025591(n); if n mod 4 = 1 or 2 then the two subsets have sums which differ by 1 and a(n) = A025591(n)/2. - Henry Bottomley, May 08 2002

More terms from Henry Bottomley, May 08 2002

Comment corrected by Steven Finch, Feb 01 2009

A060468 Number of fair distributions (equal sum) of the integers {1,..,4n} between A and B = number of solutions to the equation {+-1 +-2 +- 3 ... +-4*n = 0}.

Original entry on oeis.org

1, 2, 14, 124, 1314, 15272, 187692, 2399784, 31592878, 425363952, 5830034720, 81072032060, 1140994231458, 16221323177468, 232615054822964, 3360682669655028, 48870013251334676, 714733339229024336

Offset: 0

Roland Bacher, Mar 15 2001

			a(1)=2: give either the set {1,4} to A and {2,3} to B or give {2,3} to A and {1,4} to B.

Ray Chandler, Table of n, a(n) for n = 0..834 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A025591, A060005, A063865, A063867.

a[n_] := Coefficient[Product[q^(-k) + q^k, {k, 1, 4*n}], q, 0]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 26 2013 *)

a(n) = coefficient of q^0 in Product_{k=1..4*n} (q^(-k) + q^k).

a(n) = A025591(4n) = A063865(4n) = A063867(4n) = 2*A060005(n). Seems to be close to sqrt(3/32Pi)*16^n/sqrt(n^3 + n^2*0.6 + n*0.1385...) and sqrt(n*Pi/2)*A063074(n). - Henry Bottomley, Jul 30 2005

A359319 Maximal coefficient of (1 + x) * (1 + x^8) * (1 + x^27) * ... * (1 + x^(n^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 7, 10, 14, 18, 27, 36, 62, 95, 140, 241, 370, 607, 1014, 1646, 2751, 4863, 8260, 13909, 24870, 41671, 73936, 131257, 228204, 411128, 737620, 1292651, 2324494, 4253857, 7487549, 13710736, 25291179, 44938191, 82814603

Offset: 0

Ilya Gutkovskiy, Dec 25 2022

nonn

Conjecture: Maximal coefficient of Product_{k=1..n} (1 + x^(n^m)) ~ sqrt(4*m + 2) * 2^n / (sqrt(Pi) * n^(m + 1/2)), for m>=0. - Vaclav Kotesovec, Dec 30 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A000537, A000578, A001405, A025591, A160235, A279329, A359320.

Table[Max[CoefficientList[Product[1+x^(k^3),{k,n}],x]],{n,0,44}] (* Stefano Spezia, Dec 25 2022 *)
nmax = 100; poly = ConstantArray[0, nmax^2*(nmax + 1)^2/4 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^3 + 1]], {j, k^2*(k + 1)^2/4, k^3, -1}]; Print[k, " ", Max[poly]], {k, 2, nmax}]; (* Vaclav Kotesovec, Dec 29 2022 *)

a(n) = vecmax(Vec(prod(i=1, n, (1+x^(i^3))))); \\ Michel Marcus, Dec 27 2022

Conjecture: a(n) ~ sqrt(14) * 2^n / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Dec 30 2022

A086394 (-1) times minimal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 8, 10, 11, 12, 16, 19, 21, 23, 29, 34, 41, 46, 56, 68, 80, 92, 114, 135, 158, 182, 225, 269, 320, 369, 455, 544, 644, 753, 921, 1111, 1321, 1543, 1891, 2274, 2711, 3183, 3895, 4694, 5591, 6592, 8051, 9729, 11624

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 08 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
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]
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No.1, December 2008. pp. 176-187. [From _Johannes W. Meijer_, Jun 21 2010]

Cf. A086376.

Cf. A025591.

p:= proc(n) option remember; expand(
      `if`(n=0, 1, (x^n-1)*p(n-1)))
    end:
a:= n-> -min(coeffs(p(n))):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 12 2017

p[n_] := p[n] = Expand[If[n == 0, 1, (x^n - 1)*p[n - 1]]];
a[n_] := -Min[CoefficientList[p[n], x]];
Table[a[n], {n, 1, 80}]; (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

a(n)=-vecmin(vector(n*(n+1)/2,i,polcoeff(prod(k=1,n,1-x^k),i))) \\ Benoit Cloitre, Sep 12 2003

More terms from Benoit Cloitre, Sep 12 2003

Further terms from Sascha Kurz, Sep 22 2003

cp's OEIS Frontend

A083527 a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns.

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A350457 Maximal coefficient of (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(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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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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A063867 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or +- 1.

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Mathematica

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Extensions

A039828 Largest coefficient in expansion of Product_{i=1..n} (1 + (-q)^i).

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PARI

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A069918 Number of ways of partitioning the set {1...n} into two subsets whose sums are as nearly equal as possible.

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

A086394 (-1) times minimal coefficient of the polynomial (1-x)(1-x^2)...*(1-x^n).