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A319140
Total number of binary digits in all partitions of n into distinct parts.
2
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
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
KEYWORD
nonn,base
AUTHOR
David S. Newman, Sep 11 2018
STATUS
approved