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A058377 Number of solutions to 1 +- 2 +- 3 +- ... +- n = 0. 16
0, 0, 1, 1, 0, 0, 4, 7, 0, 0, 35, 62, 0, 0, 361, 657, 0, 0, 4110, 7636, 0, 0, 49910, 93846, 0, 0, 632602, 1199892, 0, 0, 8273610, 15796439, 0, 0, 110826888, 212681976, 0, 0, 1512776590, 2915017360, 0, 0, 20965992017, 40536016030, 0, 0, 294245741167 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Consider the set { 1,2,3,...,n }. Sequence gives number of ways this set can be partitioned into 2 subsets with equal sums. For example, when n = 7, { 1,2,3,4,5,6,7} can be partitioned in 4 ways: {1,6,7} {2,3,4,5}; {2,5,7} {1,3,4,6}; {3,4,7} {1,2,5,6} and {1,2,4,7} {3,5,6}. - sorin (yamba_ro(AT)yahoo.com), Mar 24 2007

The "equal sums" of Sorin's comment are the positive terms of A074378 (Even triangular numbers halved). In the current sequence a(n) <> 0 iff n is the positive index (A014601) of an even triangular number (A014494). - Rick L. Shepherd, Feb 09 2010

a(n) is the number of partitions of n(n-3)/4 into distinct parts not exceeding n-1. - Alon Amit, Oct 18 2017

LINKS

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..3342 (terms < 10^1000, first 1000 terms from Alois P. Heinz)

Larry Glasser, A formula for A058377, Jul 29 2019

FORMULA

a(n) is half the coefficient of q^0 in product('(q^(-k)+q^k)', 'k'=1..n) for n >= 1. - Floor van Lamoen, Oct 10 2005

a(4n+1) = a(4n+2) = 0. - Michael Somos, Apr 15 2007

EXAMPLE

1+2-3=0, so a(3)=1;

1-2-3+4=0, so a(4)=1;

1+2-3+4-5-6+7=0, 1+2-3-4+5+6-7=0, 1-2+3+4-5+6-7=0, 1-2-3-4-5+6+7=0, so a(7)=4.

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-> `if`(irem(n-1, 4)<2, 0, b(n, n-1)):

seq(a(n), n=1..60);  # Alois P. Heinz, Oct 30 2011

MATHEMATICA

f[n_, s_] := f[n, s] = Which[n == 0, If[s == 0, 1, 0], Abs[s] > (n*(n + 1))/2, 0, True, f[n - 1, s - n] + f[n - 1, s + n]]; Table[ f[n, 0]/2, {n, 1, 50}]

CROSSREFS

Cf. A000217, A014601, A014494, A025591, A063865, A063866, A063867, A069918, A074378, A111133, A161943.

Column k=2 of A275714.

Sequence in context: A146294 A133982 A069179 * A023961 A147863 A019976

Adjacent sequences:  A058374 A058375 A058376 * A058378 A058379 A058380

KEYWORD

nonn

AUTHOR

Naohiro Nomoto, Dec 19 2000

EXTENSIONS

More terms from Sascha Kurz, Mar 25 2002

Edited and extended by Robert G. Wilson v, Oct 24 2002

STATUS

approved

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Last modified October 14 11:36 EDT 2019. Contains 327996 sequences. (Running on oeis4.)