A000980 -id:A000980 - cp's OEIS frontend

A047997 Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.

1, 1, 2, 1, 3, 5, 1, 4, 8, 12, 1, 5, 13, 24, 32, 1, 6, 18, 43, 73, 94, 1, 7, 25, 69, 141, 227, 289, 1, 8, 32, 104, 252, 480, 734, 910, 1, 9, 41, 150, 414, 920, 1656, 2430, 2934, 1, 10, 50, 207, 649, 1636, 3370, 5744, 8150, 9686, 1, 11, 61, 277, 967

Offset: 1

N. J. A. Sloane

Also the number of k-subsets of {1..2n-1} with mean n. - Gus Wiseman, Apr 16 2023

			From _Gus Wiseman_, Apr 18 2023: (Start)
Triangle begins:
    1
    1    2
    1    3    5
    1    4    8   12
    1    5   13   24   32
    1    6   18   43   73   94
    1    7   25   69  141  227  289
    1    8   32  104  252  480  734  910
    1    9   41  150  414  920 1656 2430 2934
Row n = 4 counts the following balanced subsets:
  {0}  {-1,1}  {-1,0,1}   {-3,0,1,2}
       {-2,2}  {-2,0,2}   {-4,0,1,3}
       {-3,3}  {-3,0,3}   {-2,-1,0,3}
       {-4,4}  {-3,1,2}   {-2,-1,1,2}
               {-4,0,4}   {-3,-1,0,4}
               {-4,1,3}   {-3,-1,1,3}
               {-2,-1,3}  {-3,-2,1,4}
               {-3,-1,4}  {-3,-2,2,3}
                          {-4,-1,1,4}
                          {-4,-1,2,3}
                          {-4,-2,2,4}
                          {-4,-3,3,4}
(End)

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Last column is a(n,n) = A002838(n).
Row sums are A212352(n) = A047653(n)-1 = A000980(n)/2-1.
A007318 counts subsets by length, A327481 by mean, A013580 by median.
A327475 counts subsets with integer mean.
Cf. A000975, A024718, A070925, A079309, A326512, A326513, A361801, A362046.

a[n_, k_] := Length[ IntegerPartitions[ n*(2k - n + 1)/2, n, Range[2k - n + 1]]]; Flatten[ Table[ a[n, k], {k, 1, 11}, {n, 1, k}]] (* Jean-François Alcover, Jan 02 2012 *)
Table[Length[Select[Subsets[Range[-n,n]],Length[#]==k&&Total[#]==0&]],{n,8},{k,n}] (* Gus Wiseman, Apr 16 2023 *)

Equivalent to number of partitions of n(2k-n+1)/2 into up to n parts each no more than 2k-n+1 so a(n, k)=A067059(n, n(2k-n+1)/2); row sums are A047653(n)-1 = A212352(n). - Henry Bottomley, Aug 11 2001

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

A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.

Original entry on oeis.org

2, 4, 6, 10, 12, 20, 24, 34, 44, 64, 78, 116, 148, 208, 286, 410, 556, 808, 1120, 1620, 2308, 3352, 4784, 6980, 10064, 14680, 21296, 31128, 45276, 66288, 96712, 141654, 207156, 303716, 444748, 652612, 956884, 1404920, 2062080, 3029564, 4450120

Offset: 1

Thomas Ward, Jan 07 2008

nonn

This quantity arises in some examples associated to the dynamical Mertens's theorem for quasihyperbolic toral automorphisms.
The function being integrated to compute a_n vanishes on the set of points in the Farey sequence of level n. I am particularly interested in knowing how large the sequence is asymptotically.
a(n) = coefficient of x^(n*(n+1)/2) in the polynomial (-1)^n*Product_{k=1..n} (1-x^k)^2, and is the maximal such coefficient as well. - Steven Finch, Feb 03 2009

			a(2) = 4 since Integral_{0..1} sin^2(Pi*x) sin^2(2*Pi*x) dx = 1/4.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000 (terms 1..174 from Robert Israel)
Miklos Bóna, R. Gómez, M. D. Ward, Workshop in Analytic and Probabilistic Combinatorics BIRS-16w5048 2016.
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]
Jeffrey Gaither, Guy Louchard, Stephan Wagner, and Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics, Combinatorics, Probability and Computing, 24 (2015), 195-215. Special Issue Honouring the Memory of Philippe Flajolet.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, arXiv:0801.2082 [math.DS], 2008-2010.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1819-1824.
Yasuhiko Kamiyama, The Euler characteristic of the fiber product of Morse functions, Bull. Korean Math. Soc. (2025) Vol. 62, No. 1, pp. 71-80. See pp. 73, 75.
Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture. Precise Asymptotic Analysis of an Integer Sequence Motivated by the Dynamical Mertens' Theorem for Quasihyperbolic Toral Automorphisms, Slides, 2013.
T. Ward, D. W. Cantrell and R. Israel, sci.math.research discussion, 2008.

Cf. A005728, A047653.

a:= n->int(product(4*(sin(Pi*j*x))^2, j=1..n), x=0..1); seq(a(n), n=1..10);
# second Maple program:
A133871:= k -> (-1)^k*coeff(mul((t^j-1)^2,j=1..k),t,k*(k+1)/2);
# Robert Israel, Mar 15 2013

p = 1; Table[p = Expand[p*(1 - x^n)^2]; Max[(-1)^n*CoefficientList[p, x]], {n, 1, 100}] (* Vaclav Kotesovec, May 03 2018 *)
(* The constant "d" *) Chop[-E^(-I*(Pi^2*(1 + 6*x^2) - 6*PolyLog[2, E^(2*I*Pi*x)]) / (6*Pi*x)) /. x -> (x /. FindRoot[Pi*(Pi*(-1 + 6*x^2) + 12*I*x*Log[1 - E^(2*I*Pi*x)]) + 6*PolyLog[2, E^(2*I*Pi*x)], {x, 4/5}, WorkingPrecision -> 100])] (* Vaclav Kotesovec, May 04 2018 *)

a(n)=sum(k=0, n*(n+1)/2, polcoeff(prod(m=1, n, 1-x^m+x*O(x^k)), k)^2) \\ Paul D. Hanna

a(n) = sum of squares of coefficients in Product_{k=1..n} (1-x^k). - Paul D. Hanna, Nov 30 2010

a(n) ~ c * d^n / sqrt(n), where d = 1.48770584269062356180051131... and c = 2.40574583936181024... [Ward, 2013]. - Vaclav Kotesovec, May 03 2018

More terms from Steven Finch, Feb 03 2009

A181765 Number of subsets of the interval [-n .. +n] with sums > 0.

Original entry on oeis.org

0, 2, 12, 54, 230, 948, 3860, 15624, 63014, 253588, 1019072, 4091174, 16412668, 65808044, 263755984, 1056789662, 4233176854, 16953418148, 67885557896, 271793651816, 1088059997732, 4355377285932, 17432688395816, 69770793302408, 279227252601884

Offset: 0

Reinhard Zumkeller

nonn

a(n) = A000302(n) - A047653(n) = (A004171(n) - A000980(n)) / 2.

			a(1) = #{{0,1}, {1}} = 2;
a(2) = #{{-2,0,1,2}, {-2,1,2}, {-1,0,1,2}, {-1,0,2}, {-1,1,2}, {-1,2}, {0,1}, {0,1,2}, {0,2}, {1}, {1,2}, {2}} = 12.

Ray Chandler, Table of n, a(n) for n = 0..1660 (terms < 10^1000)

import Data.List (subsequences)
a181765 n = length [xs | xs <- subsequences [-n..n], sum xs > 0]
-- Reinhard Zumkeller, Feb 22 2012, Nov 13 2010

A014225 Number of initial pieces needed to reach level n in the Solitaire Army game.

Original entry on oeis.org

1, 2, 4, 8, 20

Offset: 0

N. J. A. Sloane

Level 4 is the highest level that can be reached.

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 715.

Burkard Polster, How did Fibonacci beat the Solitaire army?, Mathologer video, 2022.
Zvezdelina Stankova and Brady Haran, Conway Checkers (proof), Numberphile video, 2018.
Eric Weisstein's World of Mathematics, Conway's Soldiers

Cf. A000980.

A039826 Largest coefficient in expansion of Product_{i=1..n} (1 + q^i + q^(2i)).

Original entry on oeis.org

1, 2, 3, 7, 15, 36, 87, 217, 549, 1423, 3735, 9911, 26513, 71581, 194681, 532481, 1464029, 4045117, 11225159, 31268577, 87404465, 245101771, 689323849, 1943817227, 5494808425, 15568077235, 44200775239, 125739619467, 358347118257, 1022994133467, 2925044957099, 8376049588815, 24018964753341

Offset: 1

Olivier Gérard

nonn

Ray Chandler, Table of n, a(n) for n = 1..2106 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A039827.

nmax = 28; d = {1}; a1 = {};
Do[
  n0s = Table[0, {n}];
  d = Join[d, n0s, n0s] + Join[n0s, d, n0s] + Join[n0s, n0s, d];
  AppendTo[a1, Last[Union[d]]];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 26 2014 *)

a(n)=vecmax(Vec(prod(k=1,n,1+x^k+x^(2*k))));
vector(50,n,a(n)) \\ Joerg Arndt, Jan 31 2024

a(n) ~ 3^(n+1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 11 2018

A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 13, 20, 32, 52, 90, 152, 268, 472, 845, 1520, 2766, 5044, 9277, 17112, 31724, 59008, 110162, 206260, 387282, 729096, 1375654, 2601640, 4929378, 9358944, 17797100, 33904324, 64678112, 123580884, 236413054, 452902072

Offset: 0

Kamran Reihani (reyhan_k(AT)modares.ac.ir), Jun 21 2003

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..500
K. Reihani, C*-algebras from Anzai flows and their K-groups, arXiv preprint arXiv:math/0311425 [math.OA], 2003.
K. Reihani, K-theory of Furstenberg transformation group C^*-algebras, arXiv preprint arXiv:1109.4473 [math.OA], 2011.

Cf. A000980.

A084239 := proc(n)
    local tt,c ;
    if type(n,'odd') then
        product( 1+t^(i-(n+1)/2),i=1..n) ;
    else
        (1+t^(1/2))*product( 1+t^(i-(n+1)/2),i=1..n) ;
    end if;
    tt := expand(%) ;
    for c in tt do
        if c = lcoeff(c) then
            return c ;
        end if;
    end do:
end proc: # R. J. Mathar, Nov 13 2016

a[n_] := SeriesCoefficient[If[OddQ[n], 1, 1 + Sqrt[t]]*Product[1 + t^(i - (n + 1)/2), {i, n}], {t, 0, 0}];
Array[a, 36, 0] (* Jean-François Alcover, Nov 24 2017 *)

a(n) = constant term of prod(i=1, n, 1+t^(i-.5(n+1))) for odd n and a(n) = constant term of (1+t^(.5))*prod(i=1, n, 1+t^(i-.5(n+1))) for even n.

Sums of antidiagonals of A067059, i.e. a(n) is sum over k of number of partitions of [k(n-k)/2] into up to k parts each no more than n-k. Close to 2^(n+1)*sqrt(6/(Pi*n^3)) and seems to be even closer to something like 2^(n+1)*sqrt(6/(Pi*(n^3+0.9*n^2-0.1825*n+1.5))). - Henry Bottomley, Jul 20 2003

More terms from Henry Bottomley, Jul 20 2003

A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

Original entry on oeis.org

1, 2, 4, 8, 16, 40, 88, 222, 570, 1564, 4516, 13874, 41866, 137432, 442964, 1492610, 4998674, 17204844, 59175316, 207299554, 727137516, 2582078416, 9179001124, 32943918428, 118453240846, 428937325964, 1556421977612, 5676923326262, 20754245720206

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000980, A047653, A158092, A350249, A350881.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1/x^(n^2))^2*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..28);  # Alois P. Heinz, Jan 28 2022

Table[Coefficient[Product[(x^(k^2) + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Conjecture: a(n) ~ sqrt(5) * 4^n / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Feb 05 2022

A348639 Number of ways to express n in the form 1 +- 2 +- 3 ... +- n.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 6, 11, 0, 0, 57, 103, 0, 0, 615, 1131, 0, 0, 7209, 13467, 0, 0, 89261, 168515, 0, 0, 1147893, 2183943, 0, 0, 15181540, 29055149, 0, 0, 205171534, 394497990, 0, 0, 2820847321, 5444272739, 0, 0, 39329485312, 76142226498, 0, 0, 554756557011

Offset: 1

Daniel Cortild, Oct 26 2021

nonn

a(n) is the coefficient of x^(n*(n+3)/4-1) of Product_{k=2..n} (1+x^k). - Jianing Song, Nov 19 2021

Jianing Song, Table of n, a(n) for n = 1..1000

Cf. A000980, A058377.

int solsN(int n, int k, int sum) { if (n == k) return sum == n; return solsN(n, k+1, sum + k + 1) + solsN(n, k+1, sum - k - 1);}
int getNumber(int n) { return solsN(n, 1, 1); }

list(n) = my(poly=vector(n), v=vector(n)); poly[1]=1; for(k=2, n, poly[k]=poly[k-1]*(1+'x^k)); for(k=1, n, if(k%4==2||k%4==3, v[k]=0, v[k]=polcoeff(poly[k], k*(k+3)/4-1))); v \\ Jianing Song, Nov 19 2021

from functools import cache
@cache
def b(t, s, u): # target, sum, upto
    if u == 1: return int(t == s + 1)
    return b(t, s - u, u - 1) + b(t, s + u, u - 1)
def a(n): return b(n, 0, n)
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Oct 29 2021

a(30)-a(48) from Michael S. Branicky, Oct 29 2021

A350881 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1/x^prime(k))^2.

Original entry on oeis.org

1, 2, 4, 10, 24, 50, 140, 368, 1152, 3682, 11784, 39902, 134612, 463066, 1635092, 5818384, 20684072, 73693068, 266943648, 967762792, 3533666568, 13036452946, 48102671884, 178315730764, 661567489568, 2450447537226, 9123572154720, 34201574126260

Offset: 0

Ilya Gutkovskiy, Jan 20 2022

nonn

Cf. A000980, A022894, A022896, A047653.

p:= proc(n) option remember; `if`(n=0, 1,
      p(n-1)*(x^ithprime(n)+1/x^ithprime(n))^2)
    end:
a:= n-> coeff(p(n), x, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2022

p[n_] := p[n] = If[n == 0, 1, p[n - 1]*(x^Prime[n] + 1/x^Prime[n])^2];
a[n_] := Coefficient[p[n], x, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

a(n) = polcoef (prod(k=1, n, (x^prime(k) + 1/x^prime(k))^2), 0); \\ Michel Marcus, Jan 21 2022

cp's OEIS Frontend

A047997 Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.

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

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Maple

Mathematica

PARI

Extensions

A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4*sin^2(Pi*j*x) dx.

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Examples

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Programs

Maple

Mathematica

PARI

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Extensions

A181765 Number of subsets of the interval [-n .. +n] with sums > 0.

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Programs

Haskell

A014225 Number of initial pieces needed to reach level n in the Solitaire Army game.

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Comments

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Crossrefs

A039826 Largest coefficient in expansion of Product_{i=1..n} (1 + q^i + q^(2i)).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A350495 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1/x^(k^2))^2.

Views

Author

Keywords

Links

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

A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.