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 A003405 M0754 #35 Jan 31 2024 10:39:23
%S A003405 1,1,2,3,6,8,13,19,30,41,59,80,113,149,202,264,350,447,578,730,928,
%T A003405 1155,1444,1777,2193,2667,3249,3915,4721,5635,6728,7967,9432,11083,
%U A003405 13016,15191,17717,20544,23801,27440,31604,36234,41501,47345,53954,61260,69480,78546,88699
%N A003405 G.f.: (1 + x^4 + x^7 + 2*x^8 + x^9 + x^12 + x^16)/Product_{i=1..8} (1 - x^i).
%C A003405 Enumerates certain partially ordered sets of integers.
%D A003405 J. C. P. Miller, On the enumeration of partially ordered sets of integers, pp. 109-124 of T. P. McDonough and V. C. Mavron, editors, Combinatorics: Proceedings of the Fourth British Combinatorial Conference 1973. London Mathematical Society, Lecture Note Series, Number 13, Cambridge University Press, NY, 1974. The g.f. is P(t) on page 122.
%D A003405 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003405 Ray Chandler, <a href="/A003405/b003405.txt">Table of n, a(n) for n = 0..1000</a>
%H A003405 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%H A003405 <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).
%F A003405 a(n) = p(n,8) + p(n-4,8) + p(n-7,8) + 2*p(n-8,8) + p(n-9,8) + p(n-12,8) + p(n-16,8) where p(n,k) is the number of partitions of n into at most k parts. - _Sean A. Irvine_, Apr 22 2015
%p A003405 (1+x^4+x^7+2*x^8+x^9+x^12+x^16)/mul(1-x^i,i=1..8);
%t A003405 CoefficientList[Series[(1+x^4+x^7+2x^8+x^9+x^12+x^16)/Product[1-x^i,{i,8}],{x,0,50}],x] (* or *) LinearRecurrence[{1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1},{1,1,2,3,6,8,13,19,30,41,59,80,113,149,202,264,350,447,578,730,928,1155,1444,1777,2193,2667,3249,3915,4721,5635,6728,7967,9432,11083,13016,15191},50] (* _Harvey P. Dale_, Jan 30 2024 *)
%Y A003405 Cf. A003402, A003403, A003404, A029073, A256975, A256976, A256977.
%K A003405 nonn
%O A003405 0,3
%A A003405 _N. J. A. Sloane_
%E A003405 Entry revised by _N. J. A. Sloane_, Apr 22 2015