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A343820 Number of partitions of 2n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

%I A343820 #16 Feb 13 2023 05:17:56
%S A343820 1,1,2,3,6,8,12,15,26,32,42,50,68,80,98,113,166,192,230,262,318,360,
%T A343820 418,468,572,640,732,812,934,1032,1160,1273,1626,1792,2010,2202,2482,
%U A343820 2712,3006,3268,3682,4000,4402,4762,5254,5672,6190,6658,7492,8064,8772,9412
%N A343820 Number of partitions of 2n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.
%H A343820 Alois P. Heinz, <a href="/A343820/b343820.txt">Table of n, a(n) for n = 0..10000</a>
%F A343820 a(n) is odd <=> n in { A000225 }.
%F A343820 a(2^(n-1)) = A002449(n).
%e A343820 a(2) = 2: [1,1,1,1], [1,1,2].
%e A343820 a(3) = 3: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
%e A343820 a(4) = 6: [1,1,1,1,1,1,1,1], [1,1,1,1,1,1,2], [1,1,1,1,2,2], [1,1,2,2,2], [1,1,1,1,4], [1,1,2,4].
%p A343820 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0, (p->
%p A343820       `if`(p>n or p>n-p+1, 0, b(n-p, i)))(2^i)+b(n, i-1)))
%p A343820     end:
%p A343820 a:= n-> b(2*n, ilog2(n)+1):
%p A343820 seq(a(n), n=0..80);
%t A343820 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 0, 0, Function[p, If[p > n || p > n - p + 1, 0, b[n - p, i]]][2^i] + b[n, i - 1]]];
%t A343820 a[n_] := b[2n, BitLength[n] + 1];
%t A343820 Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Feb 13 2023, after _Alois P. Heinz_ *)
%Y A343820 Cf. A000123, A000225, A002449, A018819, A343756, A343944.
%K A343820 nonn
%O A343820 0,3
%A A343820 _Alois P. Heinz_, Apr 30 2021