A343944 -id:A343944 - cp's OEIS frontend

A281688 Expansion of Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).

1, 3, 5, 10, 14, 23, 29, 45, 55, 79, 93, 130, 150, 199, 225, 296, 332, 423, 469, 594, 654, 807, 881, 1085, 1179, 1423, 1537, 1850, 1990, 2355, 2521, 2983, 3185, 3719, 3957, 4618, 4902, 5655, 5985, 6909, 7299, 8343, 8793, 10050, 10574, 11979, 12577, 14260, 14952, 16823, 17609, 19818, 20718, 23155, 24169, 27033

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into powers of 2 (A000079).

Convolution of A001511 and A018819.

			a(4) = 10 because we have [4], [2, 2], [2, 1, 1], [1, 1, 1, 1] and 1 + 2 + 3 + 4 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for related partition-counting sequences

Cf. A000079, A001511, A018819, A343944.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p->
      `if`(p>n, 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=1..56);  # Alois P. Heinz, May 04 2021

Rest[CoefficientList[Series[Sum[x^2^i/(1 - x^2^i), {i, 0, 20}]/Product[1 - x^2^j, {j, 0, 20}], {x, 0, 56}], x]]

G.f.: Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).

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.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 12, 15, 26, 32, 42, 50, 68, 80, 98, 113, 166, 192, 230, 262, 318, 360, 418, 468, 572, 640, 732, 812, 934, 1032, 1160, 1273, 1626, 1792, 2010, 2202, 2482, 2712, 3006, 3268, 3682, 4000, 4402, 4762, 5254, 5672, 6190, 6658, 7492, 8064, 8772, 9412

Offset: 0

Alois P. Heinz, Apr 30 2021

nonn

			a(2) = 2: [1,1,1,1], [1,1,2].
a(3) = 3: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
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].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000123, A000225, A002449, A018819, A343756, A343944.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0, (p->
      `if`(p>n or p>n-p+1, 0, b(n-p, i)))(2^i)+b(n, i-1)))
    end:
a:= n-> b(2*n, ilog2(n)+1):
seq(a(n), n=0..80);

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]]];
a[n_] := b[2n, BitLength[n] + 1];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

a(n) is odd <=> n in { A000225 }.

a(2^(n-1)) = A002449(n).

cp's OEIS Frontend

A281688 Expansion of Sum_{i>=0} x^(2^i)/(1 - x^(2^i)) / Product_{j>=0} (1 - x^(2^j)).

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