A319140 Total number of binary digits in all partitions of n into distinct parts.

1, 2, 5, 6, 11, 17, 23, 30, 44, 60, 76, 102, 128, 166, 214, 264, 327, 413, 502, 618, 759, 917, 1105, 1335, 1598, 1907, 2279, 2702, 3191, 3776, 4436, 5198, 6101, 7113, 8292, 9653, 11188, 12951, 14984, 17277, 19889, 22881, 26248, 30073, 34439, 39320, 44850

Offset: 1

Author

David S. Newman, Sep 11 2018

Keywords

• nonn
• base

Examples

			For n = 4 there are 2 partitions into distinct parts in binary they are: 100, 11+1, for a total of 6 binary parts.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000009, A070939, A318756, A319142, A347060.

Programs

Maple

h:= proc(n) option remember; 1+ilog2(n) end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+h(i)
       *[0, p[1]])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 27 2018

Mathematica

h[n_] := h[n] = 1+Log[2, n] // Floor;
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[n > i*(i+1)/2, 0, b[n, i-1] + Function[p, p + h[i]*{0, p[[1]]}][b[n-i, Min[n-i, i-1]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[1; 60] (* Jean-François Alcover, Sep 28 2019, after Alois P. Heinz *)

PARI

seq(n)={[subst(deriv(p,y), y, 1) | p<-Vec(-1 + prod(k=1, n, 1 + x^k*y^(logint(k,2)+1) + O(x*x^n)))]} \\ Andrew Howroyd, Sep 17 2018