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

0,3

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A006128, A008284, A036037.

Sequence in context: A188589 A105082 A059821 * A182738 A192957 A094584

Adjacent sequences: A296007 A296008 A296009 * A296011 A296012 A296013

Keyword

nonn

Author

Matthew C. Russell, Dec 02 2017

Status

approved