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

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

Author

Ilya Gutkovskiy, May 20 2018

Keywords

• nonn

Comments

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

Convolution of the sequences A018819 and A038712.

Links

• Index entries for sequences related to partitions

Crossrefs

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

Programs

Mathematica

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}]

Formula

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).