A100505 Bisection of A001523.

1, 2, 8, 27, 79, 209, 512, 1183, 2604, 5504, 11240, 22277, 43003, 81098, 149769, 271404, 483439, 847681, 1464999, 2498258, 4207764, 7005688, 11538936, 18814423, 30387207, 48641220, 77205488, 121567834, 189974638, 294742961, 454164484, 695254782, 1057704607

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Maple

seq(coeff(convert(series(1+add(-(-1)^k*x^(k*(k+1)/2), k=1..100)/(mul(1-x^k, k=1..100))^2, x, 100), polynom), x, 2*n), n=0..45); # (C. Ronaldo)
# second Maple program:
b:= proc(n, i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(2*n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

Mathematica

max = 70; s = 1 + Sum[(-1)^(k+1)*q^(k*(k+1)/2), {k, 1, Sqrt[2 max] // Ceiling}]/QPochhammer[q]^2 + O[q]^max // Normal; Partition[(List @@ s) /. q -> 1, 2][[All, 1]] (* Jean-François Alcover, Apr 04 2017 *)

Prog

(Magma)
m:=200;
R<x>:=PowerSeriesRing(Integers(), m);
b:=Coefficients(R!( 1 + (&+[ x^n*(1-x^n)/(&*[(1-x^j)^2: j in [1..n]]): n in [1..m+2]]) ));
A100505:= func< n | b[2*n+1] >;
[A100505(n): n in [0..80]]; // G. C. Greubel, Apr 03 2023
(SageMath)
@CachedFunction
def b(n, k): # Indranil Ghosh's code of A001523
    if k>n: return 0
    if n%k==0: x=1
    else: x=0
    return x + sum(b(n-k*j, k+1)*(j+1) for j in range(n//k + 1))
def A100505(n): return 1 if n==0 else b(2*n, 1)
[A100505(n) for n in range(81)] # G. C. Greubel, Apr 03 2023

Crossrefs

Cf. A001523, A100506.

Sequence in context: A092071 A289864 A290056 * A102759 A292698 A261056

Adjacent sequences: A100502 A100503 A100504 * A100506 A100507 A100508

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 24 2004

Extensions

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

Status

approved