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A060005 Number of ways of partitioning the integers {1,2,..,4n} into two (unordered) sets such that the sums of parts are equal in both sets (parts in either set will add up to (4n)*(4n+1)/4). Number of solutions to {1 +- 2 +- 3 +- ... +- 4n=0}. 9
1, 1, 7, 62, 657, 7636, 93846, 1199892, 15796439, 212681976, 2915017360, 40536016030, 570497115729, 8110661588734, 116307527411482, 1680341334827514, 24435006625667338, 357366669614512168, 5253165510907071170 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..835 (terms < 10^1000, first 251 terms from Alois P. Heinz)

Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.

FORMULA

a(0)=1 and a(n) is half the coefficient of q^0 in product((q^(-k)+q^k), k=1..4*n) for n >= 1.

For n>=1, a(n) = (1/Pi)*16^n*J(4n) where J(n) = integral(t=0, Pi/2, cos(t)cos(2t)...cos(nt)dt). - Benoit Cloitre, Sep 24 2006

EXAMPLE

a(1)=1 since there is only one way of partitioning {1,2,3,4} into two sets of equal sum, namely {1,4}, {2,3}.

MAPLE

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-> b(4*n, 4*n-1):

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 30 2011

MATHEMATICA

b[n_, i_] := b[n, i] = Module[{m = i*(i+1)/2}, If[n > m, 0, If[n == m, 1, b[Abs[n-i], i-1] + b[n+i, i-1]]]]; a[n_] := b[4*n, 4*n-1]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Sep 26 2013, translated from Alois P. Heinz's Maple program *)

CROSSREFS

Cf. A060468, A007219, A107350, a(n)=A058377(4n), A227850.

Sequence in context: A327588 A287481 A289212 * A055066 A216534 A167550

Adjacent sequences:  A060002 A060003 A060004 * A060006 A060007 A060008

KEYWORD

nonn

AUTHOR

Roland Bacher, Mar 15 2001

EXTENSIONS

More terms from Alois P. Heinz, Oct 30 2011

STATUS

approved

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Last modified October 1 03:24 EDT 2020. Contains 337441 sequences. (Running on oeis4.)