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

0,3

Links

Table of n, a(n) for n=0..18.

Example

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).

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 ] ]

Crossrefs

Cf. A000041, A000110, A000217, A066740.

Sequence in context: A370886 A135335 A336989 * A000994 A266546 A271272

Adjacent sequences: A066720 A066721 A066722 * A066724 A066725 A066726

Keyword

more,nonn

Author

Naohiro Nomoto, Jan 15 2002

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

Status

approved