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A063865 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0. 28
1, 0, 0, 2, 2, 0, 0, 8, 14, 0, 0, 70, 124, 0, 0, 722, 1314, 0, 0, 8220, 15272, 0, 0, 99820, 187692, 0, 0, 1265204, 2399784, 0, 0, 16547220, 31592878, 0, 0, 221653776, 425363952, 0, 0, 3025553180, 5830034720, 0, 0, 41931984034, 81072032060, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of sum partitions of the half of the n-th-triangular number by distinct numbers in the range 1 to n. Example: a(7)=8 since, Triangular(7)=28 and 14=2+3+4+5=1+3+4+6=1+2+5+6=3+5+6=7+1+2+4=7+3+4=7+2+5=7+1+6 . - Hieronymus Fischer, Oct 20 2010

The asymptotic formula below was stated as a conjecture by Andrica & Tomescu in 2002 and proved by B. D. Sullivan in 2013. See his paper and H.-K. Hwang's review MR 2003j:05005 of the JIS paper. - Jonathan Sondow, Nov 11 2013

LINKS

T. D. Noe, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 101 terms from T. D. Noe, next 300 terms from N. J. A. Sloane)

D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.

D. Andrica and I. Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Seq., 5 (2002), Article 02.2.4

S. R. Finch, Signum equations and extremal coefficients.

B. D. Sullivan, On a Conjecture of Andrica and Tomescu, J. Int. Sequences, 16 (2013), Article 13.3.1.

zbMATH, Review of Andrica and Tomescu

FORMULA

Asymptotic formula: a(n) ~ sqrt(6/Pi)*n^(-3/2)*2^n for n = 0 or 3 (mod 4) as n approaches infinity.

a(n) = 0 unless n == 0 or 3 (mod 4).

a(n) = constant term in expansion of Prod_{ k = 1..n } (x^k + 1/x^k). - N. J. A. Sloane, Jul 07 2008

If n = 0 or 3 (mod 4) then a(n) = coefficient of x^(n(n+1)/4) in Product_{k=1..n} (1+x^k). - D. Andrica and I. Tomescu.

MAPLE

M:=400; t1:=1; lprint(0, 1); for n from 1 to M do t1:=expand(t1*(x^n+1/x^n)); lprint(n, coeff(t1, x, 0)); od: # N. J. A. Sloane, Jul 07 2008

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]]; a[n_] := f[n, 0]

nmax = 50; 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];

  , {n, nmax}];

a1 (* Ray Chandler, Mar 13 2014 *)

PROG

(PARI) a(n)=my(x='x); polcoeff(prod(k=1, n, x^k+x^-k)+O(x), 0) \\ Charles R Greathouse IV, May 18 2015

(PARI) a(n)=0^n+floor(prod(k=1, n, 2^(n*k)+2^(-n*k)))%(2^n) \\ Tani Akinari, Mar 09 2016

CROSSREFS

Cf. A000980, A025591, A058377, A063866, A063867, A113036, A113037, A141000.

"Decimations": A060468 = 2*A060005, A123117 = 2*A104456.

Analogous sequences for sums of squares and cubes are A158092, A158118, see also A019568. - Pietro Majer, Mar 15 2009

Sequence in context: A175917 A069971 A167291 * A230275 A230592 A282699

Adjacent sequences:  A063862 A063863 A063864 * A063866 A063867 A063868

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, suggested by J. H. Conway, Aug 27 2001

EXTENSIONS

More terms from Dean Hickerson, Aug 28, 2001

Corrected and edited by Steven Finch, Feb 01 2009

STATUS

approved

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Last modified March 26 18:46 EDT 2017. Contains 284137 sequences.