A318756 -id:A318756 - cp's OEIS frontend

A347060 Total number of 1's in the binary expansion of parts in all partitions of n into distinct parts.

0, 1, 1, 4, 4, 7, 11, 15, 20, 28, 39, 48, 64, 80, 104, 134, 167, 203, 257, 311, 381, 470, 566, 680, 820, 981, 1168, 1394, 1650, 1946, 2300, 2700, 3161, 3705, 4315, 5026, 5845, 6769, 7827, 9049, 10424, 11992, 13784, 15801, 18088, 20702, 23620, 26922, 30665

Offset: 0

Alois P. Heinz, Aug 14 2021

			a(5) = 7 counts the 1's in [101], [100, 1], [11, 10].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A000120, A066189, A066624, A318756, A319140.

h:= proc(n) option remember; add(i, i=Bits[Split](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+
       [0, p[1]*h(i)])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);

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

Original entry on oeis.org

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

David S. Newman, Sep 11 2018

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

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

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

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

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 *)

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

cp's OEIS Frontend

A347060 Total number of 1's in the binary expansion of parts in all partitions of n into distinct parts.

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