This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066723 #15 Nov 04 2023 02:48:18
%S A066723 1,1,2,5,13,36,109,340,1116,3744,12981,45722,165247,603242,2242932,
%T A066723 8422438,32040585,122800802,475937009
%N A066723 Number of distinct partitions of n-th triangular number which can be obtained by merging parts in the partition 1+2+...+n.
%e A066723 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).
%p A066723 b:= proc(n) b(n):= `if`(n<2, {[1$n]}, map(x-> [sort([x[], n]),
%p A066723 seq(sort(subsop(i=x[i]+n, x)), i=1..nops(x))][], b(n-1)))
%p A066723 end:
%p A066723 a:= n-> nops(b(n)):
%p A066723 seq(a(n), n=0..10); # _Alois P. Heinz_, May 31 2013
%t A066723 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 ] ]
%Y A066723 Cf. A000041, A000110, A000217, A066740.
%K A066723 more,nonn
%O A066723 0,3
%A A066723 _Naohiro Nomoto_, Jan 15 2002
%E A066723 Edited by _Dean Hickerson_, Jan 18 2002
%E A066723 a(15) from _Alois P. Heinz_, May 31 2013
%E A066723 a(16)-a(18) from _Sean A. Irvine_, Nov 03 2023