A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).

1, 2, 4, 10, 26, 76, 236, 760, 2522, 8556, 29504, 103130, 364548, 1300820, 4679472, 16952162, 61790442, 226451036, 833918840, 3084255128, 11451630044, 42669225172, 159497648600, 597950875256, 2247724108772, 8470205600640, 31991616634296, 121086752349064

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Or, constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^2. - N. J. A. Sloane, Jul 09 2008

Or, maximal coefficient of the polynomial (1+x)^2 * (1+x^2)^2 *...* (1+x^n)^2.

a(n) = A000302(n) - A181765(n).

From Gus Wiseman, Apr 18 2023: (Start)

Also the number of subsets of {1..2n} that are empty or have mean n. The a(0) = 1 through a(3) = 10 subsets are:

{} {} {} {}

{1} {2} {3}

{1,3} {1,5}

{1,2,3} {2,4}

{1,2,6}

{1,3,5}

{2,3,4}

{1,2,3,6}

{1,2,4,5}

{1,2,3,4,5}

Also the number of subsets of {-n..n} with no 0's but with sum 0. The a(0) = 1 through a(3) = 10 subsets are:

{} {} {} {}

{-1,1} {-1,1} {-1,1}

{-2,2} {-2,2}

{-2,-1,1,2} {-3,3}

{-3,1,2}

{-2,-1,3}

{-2,-1,1,2}

{-3,-1,1,3}

{-3,-2,2,3}

{-3,-2,-1,1,2,3}

(End)

Links

• T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..1669 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
• Ovidiu Bagdasar and Dorin Andrica, New results and conjectures on 2-partitions of multisets, 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).
• Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
• R. C. Entringer, Representation of m as Sum_{k=-n..n} epsilon_k k, Canad. Math. Bull., 11 (1968), 289-293.
• Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
• R. P. Stanley, Weyl groups, the hard Lefschetz theorem and the Sperner property, SIAM J. Algebraic and Discrete Methods 1 (1980), 168-184.

Crossrefs

Cf. A025591.

Cf. A053632; variant: A127728.

For median instead of mean we have A079309(n) + 1.

Odd bisection of A133406.

A000980 counts nonempty subsets of {1..2n-1} with mean n.

A007318 counts subsets by length, A327481 by mean.

Cf. A024718, A047997, A057552, A070925, A212352, A326512, A326513, A327475, A361866, A362046.

Programs

Maple

f:=n->coeff( expand( mul((x^k+1/x^k)^2,k=1..n) ),x,0);
# second Maple program:
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> b(0, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014

Mathematica

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[i == 0, 1, 2*b[n, i-1]+b[n+i, i-1]+b[Abs[n-i], i-1]]]; a[n_] := b[0, n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 26; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2];
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n] +
    2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
, {n, nmax}];
a1 (* Ray Chandler, Mar 15 2014 *)
Table[Length[Select[Subsets[Range[2n]],Length[#]==0||Mean[#]==n&]],{n,0,6}] (* Gus Wiseman, Apr 18 2023 *)

PARI

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

PARI

{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, Nov 30 2010

Formula

Sum of squares of coefficients in Product_{k=1..n} (1+x^k):

a(n) = Sum_{k=0..n(n+1)/2} A053632(n,k)^2. - Paul D. Hanna, Nov 30 2010

a(n) = A000980(n)/2.

a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2014

From Gus Wiseman, Apr 18 2023 (Start)

a(n) = A133406(2n+1).

a(n) = A212352(n) + 1.

a(n) = A362046(2n) + 1.

(End)

Extensions

More terms from Michael Somos, Jun 10 2000