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

1,2

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A103035 A088273 A049054 * A336902 A373243 A135476

Adjacent sequences: A319137 A319138 A319139 * A319141 A319142 A319143

Keyword

nonn,base

Author

David S. Newman, Sep 11 2018

Status

approved