A227850 -id:A227850 - cp's OEIS frontend

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}.

1, 1, 7, 62, 657, 7636, 93846, 1199892, 15796439, 212681976, 2915017360, 40536016030, 570497115729, 8110661588734, 116307527411482, 1680341334827514, 24435006625667338, 357366669614512168, 5253165510907071170

Offset: 0

Roland Bacher, Mar 15 2001

nonn

			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}.

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.

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

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

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-François Alcover, Sep 26 2013, translated from Alois P. Heinz's Maple program *)

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

More terms from Alois P. Heinz, Oct 30 2011

A168238 Number of different 0-moment rowing configurations for 4n rowers.

Original entry on oeis.org

1, 4, 29, 263, 2724, 30554, 361677, 4454273, 56546511, 735298671, 9749613914, 131377492010, 1794546880363, 24798396567242, 346130144084641, 4873560434459530, 69149450121948083, 987844051312409668, 14198028410251734447, 205181815270346718199

Offset: 1

Jeffrey Shallit, Nov 21 2009

nonn

Also the number of ways to assign 2n +1's and 2n -1's to the numbers 1,2,...,4n such that the sum is 0, assuming 1 gets the sign +1.

			For n = 2 there are 4 solutions to the 8-man rowing problem.
For n=1 the unique solution is 1+4 = 2+3. For n=2 there are 4 different solutions: 1+2+7+8 = 3+4+5+6, 1+3+6+8 = 2+4+5+7, 1+4+5+8 = 2+3+6+7, 1+4+6+7 = 2+3+5+8. - _Michel Marcus_, May 25 2013

Alois P. Heinz, Table of n, a(n) for n = 1..50
John D. Barrow, Rowing and the Same-Sum Problem Have Their Moments, arXiv:0911.3551 [physics.pop-ph], 2009-2010.

Bisection of row n=2 of A203986. - Alois P. Heinz, Jan 09 2012

Cf. A227850.

b[L_List] := b[L] = Module[{nL = Length[L], k = L[[-1]], m = L[[-2]]}, Which[L[[1]] == 0, If[nL == 3, 1, b[L[[2 ;; nL]]]], L[[1]] < 1, 0, True, Sum[If[L[[j]] < m, 0, b[Join[Sort[Table[L[[i]] - If[i == j, m + 1/97, 0], {i, 1, nL - 2}]], {m - 1, k}]]], {j, 1, nL - 2}]]];
A[n_, k_] := If[n==0 || k==0, 1, b[Join[Array[(k (n k + 1)/2 + k/97)&, n], {k n, k}]]/n!];
a[n_] := A[2, 2n];
Array[a, 20] (* Jean-François Alcover, Aug 19 2018, after Alois P. Heinz *)

a(6)-a(20) from Alois P. Heinz, Jan 09 2012

A073410 Number of permutations p of (1,2,3,...,n) such that 1(-1)^p(1)+2(-1)^p(2)+3(-1)^p(3)+...+n(-1)^p(n)=0.

Original entry on oeis.org

1, 0, 0, 2, 8, 0, 0, 576, 4608, 0, 0, 2505600, 30067200, 0, 0, 53444966400, 855119462400, 0, 0, 3587014803456000, 71740296069120000, 0, 0, 584198928937451520000, 14020774294498836480000, 0, 0, 196340349691596912721920000, 5497529791364713556213760000, 0, 0

Offset: 0

Benoit Cloitre, Aug 23 2002

nonn

Equivalently the number of grand Dyck n-paths in which each run length is selected from {1..2*n} without replacement. - David Scambler, Apr 16 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A227850.

b:= proc(n, i, c) option remember; `if`(abs(n)>i*(i+1)/2, 0,
      `if`(i=0, `if`(abs(c)<2, 1, 0),
       b(n+i, i-1, c+1) +b(n-i, i-1, c-1)))
    end:
a:= n-> b(0, n, 0)*floor(n/2)!*ceil(n/2)!/2^irem(n, 2):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 29 2015

b[n_, i_, c_] := b[n, i, c] = If[Abs[n] > i*(i+1)/2, 0, If[i == 0, If[Abs[c]<2, 1, 0], b[n+i, i-1, c+1] + b[n-i, i-1, c-1]]]; a[n_] := b[0, n, 0]*Floor[n/2]! *Ceiling[n/2]!/2^Mod[n, 2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 12 2015, after Alois P. Heinz *)

a(n)=sum(k=1,n!,if(sum(i=1,n,i*(-1)^component(numtoperm(n,k),i)),0,1))

It seems that a(n)=0 if n==1 or 2 (mod 4) and a(4*k)=4*k*a(4*k-1). - Benoit Cloitre, Aug 23 2002

More terms from John W. Layman, Feb 05 2003
a(14)-a(22) from Robert Gerbicz, Nov 22 2010
a(0), a(23)-a(30) from Alois P. Heinz, Apr 28 2015

cp's OEIS Frontend

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}.

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A168238 Number of different 0-moment rowing configurations for 4n rowers.

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A073410 Number of permutations p of (1,2,3,...,n) such that 1(-1)^p(1)+2(-1)^p(2)+3(-1)^p(3)+...+n(-1)^p(n)=0.

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cp's OEIS Frontend

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}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A168238 Number of different 0-moment rowing configurations for 4n rowers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A073410 Number of permutations p of (1,2,3,...,n) such that 1*(-1)^p(1)+2*(-1)^p(2)+3*(-1)^p(3)+...+n*(-1)^p(n)=0.

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Author

Keywords

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A073410 Number of permutations p of (1,2,3,...,n) such that 1(-1)^p(1)+2(-1)^p(2)+3(-1)^p(3)+...+n(-1)^p(n)=0.