A296010 Sum of the squares of the number of parts in all partitions of n.

0, 1, 5, 14, 34, 68, 133, 232, 402, 652, 1048, 1609, 2465, 3640, 5358, 7694, 10993, 15399, 21498, 29520, 40394, 54572, 73425, 97756, 129710, 170525, 223428, 290552, 376551, 484819, 622317, 794167, 1010515, 1279376, 1615126, 2029948, 2544600, 3176856, 3956277

Offset: 0

Author

Matthew C. Russell, Dec 02 2017

Keywords

• nonn

Examples

			For n=4, the 5 partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. These have 1, 2, 2, 3, and 4 parts, respectively. The sum of the squares is 1+4+4+9+16=34.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms 0..115 from Robert G. Wilson v)

Crossrefs

Cf. A006128, A008284, A036037.

Programs

Maple

K:=[]:
for n from 0 to 20 do
co:=0:
for L in combinat[partition](n) do
co:=co+nops(L)^2:
od:
K:=[op(K),co]:
od:
K;
# second Maple program:
b:= proc(n, i, c) option remember; `if`(n=0 or i=1,
      (n+c)^2, `if`(i>n, 0, b(n-i, i, c+1))+b(n, i-1, c))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 02 2017

Mathematica

f[n_] := Sum[i^2 (Length@ IntegerPartitions[n, {i}]), {i, n}]; Array[f, 34, 0] (* Robert G. Wilson v, Dec 02 2017 *)
b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1,
     (n + c)^2, If[i > n, 0, b[n - i, i, c + 1]] + b[n, i - 1, c]];
a[n_] := b[n, n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

PARI

first(n)=my(x='x+O('x^(n+1)),pr=1); concat(0,Vec(sum(j=1,n,pr*=1-x^j; j^2*x^j/pr))) \\ Charles R Greathouse IV, Dec 02 2017

Formula

G.f.: Sum_{j>=1} j^2*x^j / Product_{i=1..j} (1-x^i). - Alois P. Heinz, Dec 02 2017