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 A091601 #15 Jun 07 2020 21:53:40
%S A091601 1,1,3,6,14,30,69,153,345,771,1730,3873,8682,19450,43590,97668,218864,
%T A091601 490416,1098933,2462458,5517870,12364356,27705944,62083134,139115247,
%U A091601 311727845,698516370,1565227653,3507344882,7859219406,17610851898
%N A091601 Number of compositions (ordered partitions) of n with designated summands.
%H A091601 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A091601 INVERT(DCONV(A000012, iINVERT(A000027)))
%F A091601 G.f.: 1/(1 - sum(k>0, x^k/(1-x^k+x^(2*k)))). - _Vladeta Jovovic_, Dec 04 2004
%e A091601 a(3)=6 because the compositions of 3 with designated summands are
%e A091601 3', 2'1', 1'2', 1'11, 11'1, 111'.
%e A091601 The composition 1121 corresponds to 1'12'1' and 11'2'1'.
%t A091601 Table[l = Split /@ Flatten[Permutations /@ IntegerPartitions@n, 1];
%t A091601 Total[Table[x = l[[i]]; Product[Length@x[[j]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 15}] (* _Robert Price_, Jun 07 2020 *)
%Y A091601 Cf. A077285.
%K A091601 nonn
%O A091601 0,3
%A A091601 _Christian G. Bower_, Jan 23 2004