A085360 - OEIS

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A085360

Partial sums of A026905; the convolution of the natural numbers with the partition function.

4

1, 4, 10, 21, 39, 68, 112, 178, 274, 412, 606, 877, 1249, 1756, 2439, 3353, 4564, 6160, 8246, 10959, 14464, 18971, 24733, 32070, 41365, 53096, 67837, 86296, 109320, 137948, 173418, 217237, 271199, 337471, 418626, 517758, 638527, 785311, 963280, 1178587, 1438477, 1751541

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Riccardo Aragona, Roberto Civino, and Norberto Gavioli, An ultimately periodic chain in the integral Lie ring of partitions, J. Algebr. Comb. (2024). See p. 11.

Jon Perry, More Partition Function

FORMULA

a(n) = A086716(n) - A086716(n-1). - Vaclav Kotesovec, Jun 23 2015

a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - Vaclav Kotesovec, Jun 23 2015

a(n) = A014153(n) - (n + 1). - Andrew Howroyd, Oct 29 2025

EXAMPLE

a(4) = A026905(1) + A026905(2) + A026905(3) + A026905(4) = 1 + 3 + 6 + 11 = 21.

MATHEMATICA

s1=s2=0; lst={}; Do[AppendTo[lst, s2+=s1+=PartitionsP[n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 16 2009 *)

Nest[Accumulate, PartitionsP[Range[45]], 2] (* Harvey P. Dale, May 12 2026 *)

PROG

(PARI) seq(n)=Vec(sum(k=1, n, numbpart(k)*x^k, O(x*x^n))/(1-x)^2) \\ Andrew Howroyd, Oct 29 2025

CROSSREFS

Cf. A014153, A026905, A086716.

Sequence in context: A055908 A325951 A023538 * A265050 A376712 A243738

Adjacent sequences: A085357 A085358 A085359 * A085361 A085362 A085363

KEYWORD

nonn,changed

AUTHOR

Jon Perry, Jun 25 2003

STATUS

approved