A066723 Number of distinct partitions of n-th triangular number which can be obtained by merging parts in the partition 1+2+...+n.
1, 1, 2, 5, 13, 36, 109, 340, 1116, 3744, 12981, 45722, 165247, 603242, 2242932, 8422438, 32040585, 122800802, 475937009
Offset: 0
Examples
For n=4, the 13 partitions are 10, 1+9, 2+8, 3+7, 4+6, 5+5, 1+2+7, 1+3+6, 1+4+5, 2+3+5, 2+4+4, 3+3+4, 1+2+3+4. 3+7 and 4+6 can be obtained in two ways each: 3+7 = (3)+(1+2+4) = (1+2)+(3+4), 4+6 = (4)+(1+2+3) = (1+3)+(2+4).
Programs
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Maple
b:= proc(n) b(n):= `if`(n<2, {[1$n]}, map(x-> [sort([x[], n]), seq(sort(subsop(i=x[i]+n, x)), i=1..nops(x))][], b(n-1))) end: a:= n-> nops(b(n)): seq(a(n), n=0..10); # Alois P. Heinz, May 31 2013 -
Mathematica
addto[ p_, k_ ] := Module[ {}, lth=Length[ p ]; Union[ Sort/@Append[ Table[ Join[ Take[ p, i-1 ], {p[ [ i ] ]+k}, Take[ p, i-lth ] ], {i, 1, lth} ], Append[ p, k ] ] ] ]; addtolist[ plist_, k_ ] := Union[ Join@@(addto[ #, k ]&/@plist) ]; l[ 0 ]={{}}; l[ n_ ] := l[ n ]=addtolist[ l[ n-1 ], n ]; a[ n_ ] := Length[ l[ n ] ]
Extensions
Edited by Dean Hickerson, Jan 18 2002
a(15) from Alois P. Heinz, May 31 2013
a(16)-a(18) from Sean A. Irvine, Nov 03 2023