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
Keywords
Examples
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
Programs
-
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 *)