A058498 Number of solutions to c(1)t(1) + ... + c(n)t(n) = 0, where c(i) = +-1 for i>1, c(1) = t(1) = 1, t(i) = triangular numbers (A000217).

0, 0, 0, 1, 0, 1, 1, 2, 0, 6, 8, 13, 0, 33, 52, 105, 0, 310, 485, 874, 0, 2974, 5240, 9488, 0, 30418, 55715, 104730, 0, 352467, 642418, 1193879, 0, 4165910, 7762907, 14493951, 0, 50621491, 95133799, 179484713, 0, 637516130, 1202062094, 2273709847, 0, 8173584069

Offset

1,8

Links

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..633 (first 280 terms from Alois P. Heinz)

Formula

a(n) = [x^(n*(n+1)/2)] Product_{k=1..n-1} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)). - Ilya Gutkovskiy, Feb 01 2024

Example

a(8) = 2 because there are two solutions: 1 - 3 + 6 + 10 + 15 - 21 + 28 - 36 = 1 - 3 - 6 + 10 - 15 + 21 + 28 - 36 = 0.

Maple

b:= proc(n, i) option remember; local m; m:= (2+(3+i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i*(i+1)/2), i-1) +b(n+i*(i+1)/2, i-1)))
    end:
a:= n-> `if`(irem(n, 4)=1, 0, b(n*(n+1)/2, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 31 2011

Mathematica

b[n_, i_] := b[n, i] = With[{m = (2+(3+i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[Abs[n - i*(i+1)/2], i-1] + b[n + i*(i+1)/2, i-1]]]]; a[n_] := If[Mod[n, 4] == 1, 0, b[n*(n+1)/2, n-1]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 30 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000217.

Sequence in context: A086777 A355978 A334349 * A358167 A348189 A003076

Adjacent sequences: A058495 A058496 A058497 * A058499 A058500 A058501

Keyword

nonn

Author

Naohiro Nomoto, Dec 20 2000

Extensions

More terms from Sascha Kurz, Oct 13 2001

More terms from Alois P. Heinz, Oct 31 2011

Status

approved