A100506 - OEIS

%I #19 Apr 03 2023 14:25:58

%S 1,4,15,47,130,330,784,1765,3804,7898,15880,31048,59220,110484,202070,

%T 362974,641368,1116325,1916184,3247088,5436972,9002752,14752316,

%U 23938188,38487496,61344055,96974176,152110204,236837795,366177506,562373990,858193804,1301654610

%N Bisection of A001523.

%H Alois P. Heinz, <a href="/A100506/b100506.txt">Table of n, a(n) for n = 0..5000</a>

%p seq(coeff(convert(series(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+1),n=0..45); # (C. Ronaldo)

%p # second Maple program:

%p b:= proc(n, i) option remember;

%p       `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+

%p       add(b(n-i*j, i+1)*(j+1), j=0..n/i))

%p     end:

%p a:= n-> `if`(n=0, 1, b(2*n+1, 1)):

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 26 2014

%t b[n_, i_]:= b[n, i]= If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i + 1]*(j+1), {j, 0, n/i}]];

%t a[n_]:= If[n==0, 1, b[2*n+1, 1]];

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Jun 18 2018, after _Alois P. Heinz_ *)

%o (Magma)

%o m:=200;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o b:= Coefficients(R!( 1 + (&+[ x^n*(1-x^n)/(&*[(1-x^j)^2: j in [1..n]]): n in [1..m+2]]) ));

%o A100506:= func< n | b[2*n+2] >;

%o [A100506(n): n in [0..80]]; // _G. C. Greubel_, Apr 03 2023

%o (SageMath)

%o @CachedFunction

%o def b(n, k): # Indranil Ghosh's code of A001523

%o     if k>n: return 0

%o     if n%k==0: x=1

%o     else: x=0

%o     return x + sum(b(n-k*j, k+1)*(j+1) for j in range(n//k + 1))

%o def A100506(n): return 1 if n==0 else b(2*n+1, 1)

%o [A100506(n) for n in range(41)] # _G. C. Greubel_, Apr 03 2023

%Y Cf. A001523.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Nov 24 2004

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