A284870 Expansion of Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).

0, 1, 4, 10, 22, 42, 77, 131, 217, 345, 537, 812, 1211, 1767, 2547, 3615, 5078, 7043, 9687, 13185, 17815, 23867, 31766, 41972, 55146, 71997, 93519, 120813, 155358, 198811, 253374, 321509, 406436, 511802, 642264, 803140, 1001154, 1243966, 1541167, 1903754, 2345300, 2881404, 3531195, 4316632, 5264444, 6405389

Offset

0,3

Comments

Total number of parts in all partitions of all positive integers <= n.

Sum of largest parts of all partitions of all positive integers <= n.

From Omar E. Pol, Feb 16 2021: (Start)

Apart from initial zero this is as follows:

Convolution of A341062 and A014153.

Convolution of A000005 and A000070.

Convolution of nonzero terms of A006218 and A000041.

a(n) is also the total number of divisors of all terms in the n-th row of triangle A340581. These divisors are also all parts of all partitions of all positive integers <= n. (End)

Links

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

Index entries for related partition-counting sequences

Formula

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

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

a(n) = Sum_{k=0..n} A006128(k).

a(n) = A124920(n+1) - 1.

a(n) = Sum_{k=1..n} k * A299779(n,k). - Alois P. Heinz, May 14 2018

Example

a(4) = 22 because we have 1 = 1, 2 = 2, 1 + 1 = 2, 3 = 3, 2 + 1 = 3, 1 + 1 + 1 = 3, 4 = 4, 3 + 1 = 4, 2 + 2 = 4, 2 + 1 + 1 = 4 and 1 + 1 + 1 + 1 = 4 therefore 1 + 1 + 2 + 1 + 2 + 3 + 1 + 2 + 2 + 3 + 4 = 22 (total number of parts) or 1 + 2 + 1 + 3 + 2 + 1 + 4 + 3 + 2 + 2 + 1 = 22 (sum of largest parts).

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1) +(p-> p+[0, p[1]])(b(n-i, min(n-i, i))))
    end:
a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+b(n$2)[2]) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Feb 16 2021

Mathematica

nmax = 45; CoefficientList[Series[Sum[i x^i /(1 - x) Product[1/(1 - x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/(1 - x) Sum[x^i /(1 - x^i), {i, 1, nmax}] Product[1/(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Accumulate[Table[Sum[DivisorSigma[0, k] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]]

Crossrefs

Cf. A000070, A006128, A015724, A026905, A124920, A138137, A182738, A299779.

Cf. A000041, A006218, A014153, A340581, A341062.

Sequence in context: A227327 A023609 A055364 * A032191 A065568 A007825

Adjacent sequences: A284867 A284868 A284869 * A284871 A284872 A284873

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 04 2017

Status

approved