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

Author

N. J. A. Sloane, Nov 24 2004

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A001523, A100506.

Programs

Magma

m:=200;
R:=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

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 *)

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

Extensions

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