A003405 G.f.: (1 + x^4 + x^7 + 2*x^8 + x^9 + x^12 + x^16)/Product_{i=1..8} (1 - x^i).
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, 17717, 20544, 23801, 27440, 31604, 36234, 41501, 47345, 53954, 61260, 69480, 78546, 88699
Offset: 0
Keywords
References
- 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.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Ray Chandler, Table of n, a(n) for n = 0..1000
- Index entries for sequences related to posets
- Index entries for linear recurrences with constant coefficients, 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).
Programs
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Maple
(1+x^4+x^7+2*x^8+x^9+x^12+x^16)/mul(1-x^i,i=1..8);
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Mathematica
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 *)
Formula
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
Extensions
Entry revised by N. J. A. Sloane, Apr 22 2015
Comments