A304825 Sum of binomial(Y(2,p), 2) over the partitions p of n, where Y(2,p) is the number of part sizes with multiplicity 2 or greater in p.

1, 1, 3, 4, 9, 12, 22, 30, 50, 68, 105, 142, 210, 281, 400, 531, 736, 967, 1311, 1707, 2274, 2935, 3851, 4930, 6389, 8116, 10402, 13121, 16658, 20872, 26275, 32719, 40880, 50613, 62807, 77343, 95389, 116874, 143331, 174789, 213251, 258903, 314367, 380079, 459462

Offset

6,3

Links

Table of n, a(n) for n=6..50.

Formula

a(n) = (A301313(n) - A024788(n))/4.

G.f.: q^6 /((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).

Example

For a(8), we sum over the partitions of eight. For each partition p, we take binomial(Y(2,p),2): that is, the number of parts with multiplicity at least two choose 2.
8................B(0,2) = 0
7,1..............B(0,2) = 0
6,2..............B(0,2) = 0
6,1,1............B(1,2) = 0
5,3..............B(0,2) = 0
5,2,1............B(0,2) = 0
5,1,1,1..........B(1,2) = 0
4,4..............B(1,2) = 0
4,3,1............B(0,2) = 0
4,2,2............B(1,2) = 0
4,2,1,1..........B(1,2) = 0
4,1,1,1,1........B(1,2) = 0
3,3,2............B(1,2) = 0
3,3,1,1..........B(2,2) = 1
3,2,2,1..........B(1,2) = 0
3,2,1,1,1........B(1,2) = 0
3,1,1,1,1,1......B(1,2) = 0
2,2,2,2..........B(1,2) = 0
2,2,2,1,1........B(2,2) = 1
2,2,1,1,1,1......B(2,2) = 1
2,1,1,1,1,1,1....B(1,2) = 0
1,1,1,1,1,1,1,1..B(1,2) = 0
---------------------------
Total.....................3

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      binomial(`if`(n>1, 1, 0)+p, 2), add(
      b(n-i*j, i-1, `if`(j>1, 1, 0)+p), j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=6..60);  # Alois P. Heinz, May 19 2018

Mathematica

Array[Total[Binomial[Count[Split@#, _?(Length@# >= 2 &)], 2] & /@IntegerPartitions[#]] &, 50]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     Binomial[If[n > 1, 1, 0] + p, 2], Sum[
     b[n-i*j, i-1, If[j>1, 1, 0]+p], {j, 0, n/i}]];
a[n_] := b[n, n, 0];
a /@ Range[6, 60] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

Crossrefs

Cf. A024786, A302347.

Sequence in context: A356036 A097063 A293569 * A026476 A335604 A002513

Adjacent sequences: A304822 A304823 A304824 * A304826 A304827 A304828

Keyword

nonn

Author

Emily Anible, May 19 2018

Status

approved