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A047653 Constant term in expansion of (1/2) * Product_{k=-n..n} (1 + x^k). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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).

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.

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

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-Fran├žois 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 *)

PROG

(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

CROSSREFS

Cf. A025591.

Cf. A053632; variant: A127728.

Sequence in context: A229068 A000085 A222319 * A148100 A149815 A149816

Adjacent sequences:  A047650 A047651 A047652 * A047654 A047655 A047656

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos, Jun 10 2000

STATUS

approved

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Last modified February 23 11:41 EST 2018. Contains 299579 sequences. (Running on oeis4.)