A259399 -id:A259399 - cp's OEIS frontend

A259436 a(n) = Sum_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

1, 2, 6, 33, 658, 17465, 1789026, 172648401, 55048521937, 19738048521937, 17099936170199761, 17002207325552593617, 43456890729289136241538, 113852784934058230923022839, 667954362620824922543667163464, 4816707198961510396593071163584840

Offset: 0

Vaclav Kotesovec, Jun 27 2015

nonn

Cf. A000041, A133018, A259373, A259399, A259437, A259438, A265095.

Table[Sum[PartitionsP[k]^k,{k,0,n}],{n,0,15}]

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A265093 a(n) = Sum_{k=0..n} q(k)^2, where q(k) = partition numbers into distinct parts (A000009).

Original entry on oeis.org

1, 2, 3, 7, 11, 20, 36, 61, 97, 161, 261, 405, 630, 954, 1438, 2167, 3191, 4635, 6751, 9667, 13763, 19539, 27460, 38276, 53160, 73324, 100549, 137413, 186697, 252233, 339849, 455449, 607549, 808253, 1070397, 1412622, 1858846, 2436446, 3182942, 4147266

Offset: 0

Vaclav Kotesovec, Dec 01 2015

nonn

In general, for m >= 1, Sum_{k=0..n} q(k)^m ~ 2*sqrt(3*n)/(m*Pi) * q(n)^m ~ exp(Pi*m*sqrt(n/3)) / (Pi*m * 2^(2*m-1) * 3^(m/4-1/2) * n^(3*m/4-1/2)), where q(k) is A000009(k).

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

Cf. A000070, A036469, A259399.

Table[Sum[PartitionsQ[k]^2, {k,0,n}], {n,0,50}]

a(n) = Sum_{k=0..n} A000009(k)^2.

a(n) ~ exp(2*Pi*sqrt(n/3))/(16*Pi*n).

A209536 Number of partitions of 0 having positive part-sum <= n.

Original entry on oeis.org

1, 5, 14, 39, 88, 209, 434, 918, 1818, 3582, 6718, 12647, 22848, 41073, 72049, 125410, 213619, 361844, 601944, 995073, 1622337, 2626341, 4201366, 6681991, 10515755, 16449851, 25509951, 39333475, 60172700, 91577516, 138390480, 208096281, 310976730, 462512830

Offset: 1

Clark Kimberling, Mar 10 2012

nonn

A partition of 0 is a set {i(1), i(2),..., i(n)} of nonzero integers with sum 0. Such a set uniquely partitions into two multisets {x(1),..., x(j)} and {y(1),..., y(k)} where x(1)+x(2)+...+x(j) =-[y(1)+y(2)+...+y(k)] and x(i) > 0 and y(i) < 0 for every i. The number x(1)+x(2)+...+x(j) is the positive part-sum.

Let p(h) be the number of partitions of h>=1, as in A000041. There are p(h)^2 ways to choose each of the sets {x(1),...,x(j)} and {y(1),...,y(k)} having sum h. Consequently, there are p(1)^2+p(2)^2+...+p(n)^2 partitions of 0 having positive part-sum <= n.

			0 = 1-1 = 2-2 = 2-(1+1) = (1+1)-2 = (1+1)-(1+1),
so that a(2) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A209535, A000041, A259399.

a:= proc(n) option remember; `if`(n=0, 0,
      combinat[numbpart](n)^2+a(n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 21 2018

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]];
s[n_] := Sum[l[k]^2, {k, 1, n}];
Table[s[n], {n, 1, 40}] (* A209536 *)
(* Second program: *)
a[n_] := a[n] = If[n == 0, 0, PartitionsP[n]^2 + a[n-1]];
Array[a, 40] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

From Alois P. Heinz, Oct 21 2018: (Start)
a(n) = Sum_{j=1..n} A000041(j)^2.
a(n) = -1 + A259399(n). (End)

A259437 a(n) = Sum_{k=0..n} p(k)^n, where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 6, 37, 724, 20209, 1905630, 191250531, 57659285287, 20931112851787, 17697850924585423, 17720783665888137843, 44421728434157120665320, 117208746422032553556330253, 679595843556865572365153402674, 4907378683411420479410336076467628

Offset: 0

Vaclav Kotesovec, Jun 27 2015

nonn

Cf. A000041, A133018, A259399, A259436, A259438.

Table[Sum[PartitionsP[k]^n,{k,0,n}],{n,0,15}]

a(n) ~ p(n)^n ~ exp(1/24 - 3/(4*Pi^2) - (72+Pi^2)*sqrt(n)/(24*sqrt(6)*Pi) + sqrt(2/3)*Pi*n^(3/2)) / (3^(n/2) * 4^n * n^n).

A259438 a(n) = Sum_{k=0..n} p(k)^(n-k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 2, 3, 5, 10, 25, 78, 301, 1414, 7964, 53408, 426116, 4028890, 44697755, 576491980, 8617031811, 149425700853, 3004591733938, 69763130950599, 1860330686377532, 56746090401472922, 1975156902590115291, 78299783319570477185, 3529323014512112469681

Offset: 0

Vaclav Kotesovec, Jun 27 2015

nonn

The position of the maximum value asymptotically approaches k = n/3.

Cf. A000041, A133018, A259399, A259436, A259437.

Table[Sum[PartitionsP[k]^(n-k),{k,0,n}],{n,0,25}]

log(a(n)) ~ 2^(3/2)*Pi*n^(3/2)/9 - n*log(16*n^2/3)/3.

G.f.: Sum_{k>=0} x^k/(1 - p(k)*x). - Ilya Gutkovskiy, Oct 09 2018

cp's OEIS Frontend

A259436 a(n) = Sum_{k=0..n} p(k)^k, where p(k) is the partition function A000041.

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A265093 a(n) = Sum_{k=0..n} q(k)^2, where q(k) = partition numbers into distinct parts (A000009).

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A209536 Number of partitions of 0 having positive part-sum <= n.

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A259437 a(n) = Sum_{k=0..n} p(k)^n, where p(k) is the partition function A000041.

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Programs

Mathematica

Formula

A259438 a(n) = Sum_{k=0..n} p(k)^(n-k), where p(k) is the partition function A000041.

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Mathematica

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