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A318756 Total number of binary digits used to write all partitions of n in binary notation.

%I A318756 #24 Aug 14 2021 21:29:27
%S A318756 1,4,8,18,30,55,85,141,211,324,467,691,968,1377,1898,2631,3554,4830,
%T A318756 6425,8578,11272,14819,19243,25005,32133,41279,52585,66907,84512,
%U A318756 106636,133685,167377,208439,259145,320696,396251,487532,598881,732990,895627,1090752
%N A318756 Total number of binary digits used to write all partitions of n in binary notation.
%H A318756 Alois P. Heinz, <a href="/A318756/b318756.txt">Table of n, a(n) for n = 1..5000</a>
%e A318756 For n = 3 there are 3 partitions which when written in binary are: 11, 10+1, 1+1+1, for a total of 8 binary integers.
%p A318756 h:= proc(n) option remember; 1+ilog2(n) end:
%p A318756 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A318756       `if`(i<1, 0, b(n, i-1)+(p-> p+[0, p[1]*
%p A318756         h(i)])(b(n-i, min(n-i, i)))))
%p A318756     end:
%p A318756 a:= n-> b(n$2)[2]:
%p A318756 seq(a(n), n=1..60);  # _Alois P. Heinz_, Sep 27 2018
%t A318756 h[n_] := h[n] = 1 + Log[2, n] // Floor;
%t A318756 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, b[n, i - 1] + Function[p, p + {0, p[[1]]*h[i]}][b[n - i, Min[n - i, i]]]]];
%t A318756 a[n_] := b[n, n][[2]];
%t A318756 a /@ Range[1, 60] (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ *)
%t A318756 Table[Length[Flatten[IntegerDigits[#,2]&/@IntegerPartitions[n]]],{n,50}] (* _Harvey P. Dale_, Aug 14 2021 *)
%o A318756 (PARI) a(n)={subst(deriv(polcoef(1/prod(k=1, n, 1 - x^k*y^(logint(k,2) + 1) + O(x*x^n)), n)), y, 1)} \\ _Andrew Howroyd_, Sep 07 2018
%Y A318756 Cf. A066624, A070939.
%K A318756 nonn,base
%O A318756 1,2
%A A318756 _David S. Newman_, Sep 02 2018