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 A227682 #26 Jan 20 2015 14:08:41
%S A227682 1,1,3,7,16,35,76,162,342,715,1484,3060,6278,12824,26102,52969,107224,
%T A227682 216601,436798,879584,1769117,3554726,7136736,14318524,28711315,
%U A227682 57544864,115290624,230910993,462362571,925610398,1852669016,3707705019,7419275371,14844857959
%N A227682 G.f.: exp( Sum_{n>=1} x^n / (n*(1-x)^n * (1-x^n)) ).
%C A227682 Number of compositions of n with k sorts of parts k where the sorts of parts are nondecreasing through the composition, see example. - _Joerg Arndt_, May 01 2014
%H A227682 Alois P. Heinz, <a href="/A227682/b227682.txt">Table of n, a(n) for n = 0..500</a>
%F A227682 G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} 1/(d*(1-x)^d) ).
%F A227682 G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 35*x^5 + 76*x^6 + 162*x^7 +...
%F A227682 where
%F A227682 log(A(x)) = x/((1-x)*(1-x)) + x^2/(2*(1-x)^2*(1-x^2)) + x^3/(3*(1-x)^3*(1-x^3)) + x^4/(4*(1-x)^4*(1-x^4)) + x^5/(5*(1-x)^5*(1-x^5)) +...
%F A227682 Explicitly,
%F A227682 log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 56*x^5/5 + 107*x^6/6 + 197*x^7/7 + 365*x^8/8 + 679*x^9/9 + 1280*x^10/10 +...
%F A227682 a(n) = A238350(n*(n+3)/2,n), a(n) is the number of compositions of n*(n+3)/2 with exactly n fixed points. - _Alois P. Heinz_, Apr 11 2014
%F A227682 a(n) ~ c * 2^n, where c = 1/(2*A048651) = 1.73137330972753180576... - _Vaclav Kotesovec_, May 01 2014
%F A227682 G.f.: Product {n >= 1} 1/(1 - x^n/(1 - x)). Row sums of A253829. - _Peter Bala_, Jan 20 2015
%e A227682 From _Joerg Arndt_, May 01 2014: (Start)
%e A227682 The a(5) = 35 compositions as described in the first comment are (here p:s stands for a part p of sort s)
%e A227682 01: [ 1:0 1:0 1:0 1:0 1:0 ]
%e A227682 02: [ 1:0 1:0 1:0 2:0 ]
%e A227682 03: [ 1:0 1:0 1:0 2:1 ]
%e A227682 04: [ 1:0 1:0 2:0 1:0 ]
%e A227682 05: [ 1:0 1:0 3:0 ]
%e A227682 06: [ 1:0 1:0 3:1 ]
%e A227682 07: [ 1:0 1:0 3:2 ]
%e A227682 08: [ 1:0 2:0 1:0 1:0 ]
%e A227682 09: [ 1:0 2:0 2:0 ]
%e A227682 10: [ 1:0 2:0 2:1 ]
%e A227682 11: [ 1:0 2:1 2:1 ]
%e A227682 12: [ 1:0 3:0 1:0 ]
%e A227682 13: [ 1:0 4:0 ]
%e A227682 14: [ 1:0 4:1 ]
%e A227682 15: [ 1:0 4:2 ]
%e A227682 16: [ 1:0 4:3 ]
%e A227682 17: [ 2:0 1:0 1:0 1:0 ]
%e A227682 18: [ 2:0 1:0 2:0 ]
%e A227682 19: [ 2:0 1:0 2:1 ]
%e A227682 20: [ 2:0 2:0 1:0 ]
%e A227682 21: [ 2:0 3:0 ]
%e A227682 22: [ 2:0 3:1 ]
%e A227682 23: [ 2:0 3:2 ]
%e A227682 24: [ 2:1 3:1 ]
%e A227682 25: [ 2:1 3:2 ]
%e A227682 26: [ 3:0 1:0 1:0 ]
%e A227682 27: [ 3:0 2:0 ]
%e A227682 28: [ 3:0 2:1 ]
%e A227682 29: [ 3:1 2:1 ]
%e A227682 30: [ 4:0 1:0 ]
%e A227682 31: [ 5:0 ]
%e A227682 32: [ 5:1 ]
%e A227682 33: [ 5:2 ]
%e A227682 34: [ 5:3 ]
%e A227682 35: [ 5:4 ]
%e A227682 (End)
%t A227682 Flatten[{1,Table[SeriesCoefficient[Exp[Sum[x^k / (k*(1-x)^k * (1-x^k)),{k,1,n}]],{x,0,n}], {n,1,40}]}] (* _Vaclav Kotesovec_, May 01 2014 *)
%o A227682 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(m*(1-x)^m*(1-x^m +x*O(x^n))) )), n)}
%o A227682 for(n=0, 50, print1(a(n), ", "))
%o A227682 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, 1/(1-x +x*O(x^n))^d/d) )), n)}
%o A227682 for(n=0, 50, print1(a(n), ", "))
%Y A227682 Cf. A227681, A238349, A253829.
%K A227682 nonn
%O A227682 0,3
%A A227682 _Paul D. Hanna_, Jul 19 2013