A281688 -id:A281688 - cp's OEIS frontend

A343944 Total number of parts in all partitions of n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

0, 1, 2, 5, 7, 12, 15, 29, 35, 50, 58, 86, 98, 128, 143, 225, 251, 318, 350, 453, 495, 603, 653, 846, 914, 1092, 1172, 1419, 1517, 1773, 1886, 2521, 2687, 3130, 3322, 3917, 4147, 4759, 5021, 5909, 6227, 7082, 7442, 8537, 8955, 10076, 10544, 12326, 12898, 14452

Offset: 0

Alois P. Heinz, May 04 2021

nonn

			a(5) = 12 = 5+4+3: [1,1,1,1,1], [1,1,1,2], [1,2,2].
a(6) = 15 = 6+5+4: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
a(7) = 29 = 7+6+5+4+4+3: [1,1,1,1,1,1,1], [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,4], [1,2,4].

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A281688, A343820.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p-> `if`(
       p>n or p>n-p+1, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
    end:
a:= n-> b(n, ilog2(n))[2]:
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 0, {0, 0}, Function[p, If[p > n || p > n - p + 1, {0, 0}, Function[h, h + {0, h[[1]]}][b[n - p, i]]]][2^i] + b[n, i - 1]]];
a[n_] := b[n, Floor@Log2[n]][[2]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

A304909 Expansion of x * (d/dx) Product_{k>=0} 1/(1 - x^(2^k)).

Original entry on oeis.org

0, 1, 4, 6, 16, 20, 36, 42, 80, 90, 140, 154, 240, 260, 364, 390, 576, 612, 828, 874, 1200, 1260, 1628, 1702, 2256, 2350, 2964, 3078, 3920, 4060, 4980, 5146, 6464, 6666, 8092, 8330, 10224, 10508, 12540, 12870, 15600, 15990, 18900, 19350, 23056, 23580, 27508, 28106, 33216, 33908

Offset: 0

Ilya Gutkovskiy, May 20 2018

nonn

Sum of all parts of all partitions of n into powers of 2.

Convolution of the sequences A018819 and A038712.

Index entries for sequences related to partitions

Cf. A000079, A000123, A018819, A038712, A066186, A281688, A304908.

nmax = 49; CoefficientList[Series[x D[Product[1/(1 - x^2^k), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
nmax = 49; CoefficientList[Series[x D[Product[(1 + x^2^k)^(k + 1), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[d == 2^IntegerExponent[d, 2]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n a[n], {n, 0, 49}]

G.f.: x * (d/dx) Product_{k>=0} (1 + x^(2^k))^(k+1).
G.f.: Sum_{i>=0} 2^i*x^(2^i)/(1 - x^(2^i)) * Product_{j>=0} 1/(1 - x^(2^j)).
a(n) = n*A018819(k).

cp's OEIS Frontend

A343944 Total number of parts in all partitions of n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

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