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

%I A047653 #63 Apr 20 2023 11:42:49
%S A047653 1,2,4,10,26,76,236,760,2522,8556,29504,103130,364548,1300820,4679472,
%T A047653 16952162,61790442,226451036,833918840,3084255128,11451630044,
%U A047653 42669225172,159497648600,597950875256,2247724108772,8470205600640,31991616634296,121086752349064
%N A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k).
%C A047653 Or, constant term in expansion of Product_{k=1..n} (x^k + 1/x^k)^2. - _N. J. A. Sloane_, Jul 09 2008
%C A047653 Or, maximal coefficient of the polynomial (1+x)^2 * (1+x^2)^2 *...* (1+x^n)^2.
%C A047653 a(n) = A000302(n) - A181765(n).
%C A047653 From _Gus Wiseman_, Apr 18 2023: (Start)
%C A047653 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:
%C A047653   {}  {}   {}       {}
%C A047653       {1}  {2}      {3}
%C A047653            {1,3}    {1,5}
%C A047653            {1,2,3}  {2,4}
%C A047653                     {1,2,6}
%C A047653                     {1,3,5}
%C A047653                     {2,3,4}
%C A047653                     {1,2,3,6}
%C A047653                     {1,2,4,5}
%C A047653                     {1,2,3,4,5}
%C A047653 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:
%C A047653   {}  {}      {}           {}
%C A047653       {-1,1}  {-1,1}       {-1,1}
%C A047653               {-2,2}       {-2,2}
%C A047653               {-2,-1,1,2}  {-3,3}
%C A047653                            {-3,1,2}
%C A047653                            {-2,-1,3}
%C A047653                            {-2,-1,1,2}
%C A047653                            {-3,-1,1,3}
%C A047653                            {-3,-2,2,3}
%C A047653                            {-3,-2,-1,1,2,3}
%C A047653 (End)
%H A047653 T. D. Noe, Alois P. Heinz and Ray Chandler, <a href="/A047653/b047653.txt">Table of n, a(n) for n = 0..1669</a> (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
%H A047653 Ovidiu Bagdasar and Dorin Andrica, <a href="https://dx.doi.org/10.1109/ICMSAO.2017.7934928">New results and conjectures on 2-partitions of multisets</a>, 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).
%H A047653 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s11139-021-00418-7">On k-partitions of multisets with equal sums</a>, The Ramanujan J. (2021) Vol. 55, 421-435.
%H A047653 R. C. Entringer, <a href="http://dx.doi.org/10.4153/CMB-1968-036-3">Representation of m as Sum_{k=-n..n} epsilon_k k</a>, Canad. Math. Bull., 11 (1968), 289-293.
%H A047653 Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%H A047653 R. P. Stanley, <a href="http://dx.doi.org/10.1137/0601021">Weyl groups, the hard Lefschetz theorem and the Sperner property</a>, SIAM J. Algebraic and Discrete Methods 1 (1980), 168-184.
%F A047653 Sum of squares of coefficients in Product_{k=1..n} (1+x^k):
%F A047653 a(n) = Sum_{k=0..n(n+1)/2} A053632(n,k)^2. - _Paul D. Hanna_, Nov 30 2010
%F A047653 a(n) = A000980(n)/2.
%F A047653 a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 11 2014
%F A047653 From _Gus Wiseman_, Apr 18 2023 (Start)
%F A047653 a(n) = A133406(2n+1).
%F A047653 a(n) = A212352(n) + 1.
%F A047653 a(n) = A362046(2n) + 1.
%F A047653 (End)
%p A047653 f:=n->coeff( expand( mul((x^k+1/x^k)^2,k=1..n) ),x,0);
%p A047653 # second Maple program:
%p A047653 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
%p A047653       `if`(i=0, 1, 2*b(n, i-1)+b(n+i, i-1)+b(abs(n-i), i-1)))
%p A047653     end:
%p A047653 a:=n-> b(0, n):
%p A047653 seq(a(n), n=0..40);  # _Alois P. Heinz_, Mar 10 2014
%t A047653 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_ *)
%t A047653 nmax = 26; d = {1}; a1 = {};
%t A047653 Do[
%t A047653   i = Ceiling[Length[d]/2];
%t A047653   AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
%t A047653   d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n] +
%t A047653     2 PadLeft[PadRight[d, Length[d] + n], Length[d] + 2 n];
%t A047653 , {n, nmax}];
%t A047653 a1 (* _Ray Chandler_, Mar 15 2014 *)
%t A047653 Table[Length[Select[Subsets[Range[2n]],Length[#]==0||Mean[#]==n&]],{n,0,6}] (* _Gus Wiseman_, Apr 18 2023 *)
%o A047653 (PARI) a(n)=polcoeff(prod(k=-n,n,1+x^k),0)/2
%o A047653 (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
%Y A047653 Cf. A025591.
%Y A047653 Cf. A053632; variant: A127728.
%Y A047653 For median instead of mean we have A079309(n) + 1.
%Y A047653 Odd bisection of A133406.
%Y A047653 A000980 counts nonempty subsets of {1..2n-1} with mean n.
%Y A047653 A007318 counts subsets by length, A327481 by mean.
%Y A047653 Cf. A024718, A047997, A057552, A070925, A212352, A326512, A326513, A327475, A361866, A362046.
%K A047653 nonn
%O A047653 0,2
%A A047653 _N. J. A. Sloane_
%E A047653 More terms from _Michael Somos_, Jun 10 2000