A003405 G.f.: (1 + x^4 + x^7 + 2*x^8 + x^9 + x^12 + x^16)/Product_{i=1..8} (1 - x^i). (Formerly M0754)

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

0,3

Comments

Enumerates certain partially ordered sets of integers.

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).

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

Maple

(1+x^4+x^7+2*x^8+x^9+x^12+x^16)/mul(1-x^i, i=1..8);

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 *)

Crossrefs

Cf. A003402, A003403, A003404, A029073, A256975, A256976, A256977.

Sequence in context: A285472 A318027 A004101 * A153918 A308909 A369356

Adjacent sequences: A003402 A003403 A003404 * A003406 A003407 A003408

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Apr 22 2015

Status

approved